Annotation of ttbar/p20_taujets_note/NN.tex, revision 1.2
1.1 uid12904 1: \newpage
1.2 ! uid12904 2: %ttttt
1.1 uid12904 3: \section{\label{sub:NN}Neural Network Analysis}
4:
5: \subsection{\label{sub:Variables}Variables for NN training}
6:
7: \noindent Following the same procedure as in the previous analysis, we determine the content
8: of signal and background in the preselected sample, increase signal/background rate
9: and from this, measure the cross-section.
10: The procedure adopted in the p17 analysis was feed a set of topological variables into an
11: artificial neural network in order to provide the best possible separation between
12: signal and background. As before, the criteria for choosing such variables were: power of discrimination
13: and $\tau$-uncorrelated variables. The set is presented below:
14:
15: \begin{itemize}
16: \item \textit{\textbf{$H_{T}$}} - the scalar sum of all jet's $p_{T}$ (here and below including $\tau$ lepton candidates).
17:
18: \item \textit{\textbf{$\not\!\! E_{T}$ significance}} - As being the variable that provides the best signal-background
19: separation we decided to optimize it.
20:
21: \item \textit{\textbf{Aplanarity}} \cite{p17topo} - the normalized momentum tensor is defined as
22:
23: \begin{center}
24: \begin{equation}
25: {\cal M} = \frac{\sum_{o}p^{o}_{i}p^{o}_{j}}{\sum_{o}|\overrightarrow{p^{o}}|}
26: \label{tensor}
27: \end{equation}
28: \end{center}
29:
30: \noindent where $\overrightarrow{p^{0}}$ is the momentum-vector of a reconstructed object $o$
31: and $i$ and $j$ are cartesian coordinates. From the diagonalization of $\cal M$ we find three eigenvalues
32: $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}$ with the constraint $\lambda_{1} + \lambda_{2} + \lambda_{3} = 1$.
33: The aplanarity {\cal A} is given by {$\cal A$} = $\frac{3}{2}\lambda_{3}$ and measures the flatness of an event.
34: Hence, it is defined in the range $0 \leq {\cal M} \leq 0.5$. Large values of {$\cal A$} correspond to more spherical events,
35: like $t\bar{t}$ events for instance, since they are typical of decays of heavy objects. On the other hand,
36: both QCD and $W + \mbox{jets}$ events are more planar since jets in these events are primarily due to
37: initial state radiation.
38:
39: \item \textit{\textbf{Sphericity}} \cite{p17topo} - being defined as {$\cal S$} = $\frac{3}{2}(\lambda_{2} + \lambda_{3})$,
40: and having a range $0 \leq {\cal S} \leq 1.0$, sphericity is a measure of the summed $p^{2}_{\perp}$ with
41: respect to the event axis. In this sense a 2-jets event corresponds to {$\cal S$} $\approx 0$ and an isotropic event
42: {$\cal S$} $\approx 1$. $t\bar{t}$ events are very isotropic as they are typical of the decays of heavy objects
43: and both QCD and $W + \mbox{jets}$ events are less isotropic due to the fact that jets in these events come
44: primarily from initial state radiation.
45:
46: \item \textit{\textbf{Top and $W$ mass likelihood}} - a $\chi^{2}$-like variable.
47: $L\equiv\left(\frac{M_{3j}-m_{t}}{\sigma_{t}}\right)^{2}+\left(\frac{M_{2j}-M_{W}}{\sigma_{W}}\right)^{2}$,
48: where $m_{t}, M_{W},\sigma_{t},\sigma_{W}$ are top and W masses (172.4
49: GeV and 81.02 GeV respectively) and resolution values (19.4 GeV and 8.28
50: GeV respectively). $M_{3j}$ and $M_{2j}$ are invariant masses composed
51: of the jet combinations. We choose combination that minimizes $L$.
52:
53: \item \textit{\textbf{Centrality}}, defined as $\frac{H_{T}}{H_{E}}$ , where $H_{E}$ is sum
54: of energies of the jets.
55: \item \textit{\textbf{$\cos(\theta*)$}} - The angle between the beam axis and the
56: highest-$p_T$ jet in the rest frame of all the jets in the event.
57: \item \textit{\textbf{$\sqrt(s)$}} - The invariant mass of all jets and $\tau$s in the event.
58:
59: \end{itemize}
60:
61: The chosen variables are in the end a consequence of the method employed in this
62: analysis: use events from the QCD-enriched
63: loose-tight sample to model QCD events in the signal-rich sample, and use
64: a b-tag veto sample as an independent control sample to check the validity of such
65: background modeling.
66:
67: %\clearpage
68:
69: \subsection{\label{sub:NN-variables}Topological NN}
70: For training the Neural Network we used the Multilayer Perceptron algorithm, as described in
71: \cite{MLPfit}. As explained before in Section \ref{sub:Results-of-the}, the first 1400000
72: events in the ``loose-tight'' sample were used as background
73: for NN training for taus types 1 and 2, and the first 600000 of the same sample for NN training for type 3 taus.
74: This means that different tau types are being treated separately in the topological NN.
75: In both cases 1/3 of the Alpgen sample of $t\bar{t} \rightarrow \tau +jets$ was used for NN training and 2/3 of it
76: for the measurement.
77: When doing the measurement later on (Section \ref{sub:xsect}) we pick the tau with the highest $NN(\tau)$
78: in the signal sample as the tau cadidate at same time that taus in the loose-tight sample are picked at
79: random since all of them are regarded as fake taus by being below the cut $NN(\tau)$ = 0.7. By doing this
80: we expect to avoid any bias when selecting real taus for the measurement.
81: Figures \ref{fig:nnout_type2_training} and \ref{fig:nnout_type3_training} show the
82: effect of each of the chosen the topological event NN input variables on the final output.
83:
84: Figures \ref{fig:nnout_type2} and \ref{fig:nnout_type3} show the NN output as a result of the training
85: described above. It is evident from both pictures that high values of NN correspond to
86: the signal-enriched region.
87:
88:
89:
90: \begin{figure}[h]
91: \includegraphics[scale=0.6]{plots/SetI_NNout_SM_type2_tauQCD.eps}
92: \caption{Training of topological Neural Network output for type 1 and 2 $\tau$ channel.
93: Upper left: relative impact of each of the input variables; upper right: topological structure;
94: lower right: final signal-background separation of the method; lower left: convergence curves.}
95: \label{fig:nnout_type2_training}
96: \end{figure}
97:
98: \newpage
99:
100:
101: \begin{figure}[h]
102: \includegraphics[scale=0.6]{plots/SetI_NNout_SM_type3_tauQCD.eps}
103: \caption{Training of topological Neural Network output for type 3 $\tau$ channel.
104: Upper left: relative impact of each of the input variables; upper right: topological structure;
105: lower right: final signal-background separation of the method; lower left: convergence curves.}
106: \label{fig:nnout_type3_training}
107: \end{figure}
108:
109: \begin{figure}[h]
110: \includegraphics[scale=0.5]{CONTROLPLOTS/Std_TypeI_II/nnout.eps}
111: \caption{The topological Neural Network output for type 1 and 2 $\tau$ channel}
112: \label{fig:nnout_type2}
113: \end{figure}
114:
115: \newpage
116:
117: \begin{figure}[t]
118: \includegraphics[scale=0.5]{CONTROLPLOTS/Std_TypeIII/nnout.eps}
119: \caption{The topological Neural Network output for type 3 $\tau$ channel}
120: \label{fig:nnout_type3}
121: \end{figure}
122:
123:
124: \subsection{\label{sub:NN-optimization}NN optimization}
125: One difference between this present analysis and the previous p17 is that we performed a NN optimization along with a
126: $\not\!\! E_{T}$ significance optimization. Previously a cut of $>$ 3.0 was applied to $\not\!\! E_{T}$ significance
127: at the preselection stage and then it was included as one of the variables for NN training. This time as we
128: chose to optimize it, since it is still a good variable to provide signal-background discrimination (Figure \ref{fig:metl_note}).
129: It is important to stress out that after the optimization we performed the
130: analysis with the optimized $\not\!\! E_{T}$ significance cut
131: applied when doing both $\tau$ and b ID (Section \ref{sub:Results-of-the}), therefore
132: after the preselection where no $\not\!\! E_{T}$ significance cut was applied.
133: We then went back anp reprocessed (preselected) all MC samples with the optimized cut. Both results,
134: with $\not\!\! E_{T}$ significance applied during and after preselectio were identical.
135: We then chose to present this analyis with this cut applied at the preselection
136: level in order to have a consistent cut flow throughout the analysis(Section \ref{sub:Preselection}).
137:
138:
139: \begin{figure}[h]
140: \includegraphics[scale=0.5]{plots/metl_allEW.eps}
141: \caption{$\not\!\! E_{T}$ significance distribution for signal and backgrounds.}
142: \label{fig:metl_note}
143: \end{figure}
144:
145:
146: \newpage
147:
148: Below we describe how we split this part of the analysis into two parts:
149:
150: \begin{enumerate}
151: \item {\bf Set optimization:} We applied an arbitrary cut on $\not\!\! E_{T}$ significance of $\geq$ 4.0 and
152: varied the set of varibles going into NN training
153: \item {\bf $\not\!\! E_{T}$ significance optimization:} After chosing the best set based on the lowest RMS,
154: we then varied the $\not\!\! E_{T}$ significance cut
155: \end{enumerate}
156:
157: For this part of the analysis we present the sets of variables that were taken into account to perform the NN traning
158: \begin{itemize}
159: \item \textit{\textbf{Set I}} : {$H_{T}$}, aplan (aplanarity), sqrts ($\sqrt{s}$)
160: \item \textit{\textbf{Set II}} : {$H_{T}$}, aplan, cent (centrality)
161: \item \textit{\textbf{Set III}} : {$H_{T}$}, aplan, spher (spherecity)
162: \item \textit{\textbf{Set IV}} : {$H_{T}$}, cent, spher
163: \item \textit{\textbf{Set V}} : aplan, cent, spher
164: \item \textit{\textbf{Set VI}} : {$H_{T}$}, aplan, sqrts, spher
165: \item \textit{\textbf{Set VII}} : {$H_{T}$}, aplan, sqrts, cent
166: \item \textit{\textbf{Set VIII}} : {$H_{T}$}, aplan, sqrts, costhetastar ($cos(\theta^{*})$
167: \item \textit{\textbf{Set IX}} : {$H_{T}$}, aplan, sqrts, cent, spher
168: \item \textit{\textbf{Set X}} : {$H_{T}$}, aplan, sqrts, cent, costhetastar
169: \item \textit{\textbf{Set XI}} : {$H_{T}$}, aplan, sqrts, spher, costhetastar
170: \item \textit{\textbf{Set XII}} : metl, {$H_{T}$}, aplan, sqrts
171: \item \textit{\textbf{Set XIII}} : metl, {$H_{T}$}, aplan, cent
172: \item \textit{\textbf{Set XIV}} : metl, {$H_{T}$}, aplan, spher
173: \item \textit{\textbf{Set XV}} : metl, {$H_{T}$}, cent, spher
174: \item \textit{\textbf{Set XVI}} : metl, {$H_{T}$}, aplan
175: \item \textit{\textbf{Set XVII}} : metl, {$H_{T}$}, sqrts
176: \item \textit{\textbf{Set XVIII}} : metl, aplan, sqrts
177: \item \textit{\textbf{Set XIX}} : metl, {$H_{T}$}, cent
178: \item \textit{\textbf{Set XX}} : metl, {$H_{T}$}, aplan, sqrts, cent
179: \item \textit{\textbf{Set XXI}} : metl, {$H_{T}$}, aplan, cent, spher
180: \item \textit{\textbf{Set XXII}} : metl, {$H_{T}$}, aplan, sqrts, spher
181: \item \textit{\textbf{Set XXIII}} : metl, {$H_{T}$}, aplan, sqrts, costhetastar
182: \item \textit{\textbf{Set XXIV}} : metl, sqrts, cent, spher, costhetastar
183: \item \textit{\textbf{Set XXV}} : metl, {$H_{T}$}, cent, spher, costhetastar
184: \item \textit{\textbf{Set XXVI}} : metl, aplan, cent, spher, costhetastar
185: \item \textit{\textbf{Set XXVII}} : metl, {$H_{T}$}, aplan, cent, costhetastar
186: \item \textit{\textbf{Set XXVIII}} : {$H_{T}$}, aplan, topmassl
187: \item \textit{\textbf{Set XXIX}} : {$H_{T}$}, aplan, sqrts, topmassl
188: \item \textit{\textbf{Set XXX}} : {$H_{T}$}, aplan, sqrts, cent, topmassl
189: \item \textit{\textbf{Set XXXI}} : {$H_{T}$}, aplan, sqrts, costhetastar, topmassl
190: \item \textit{\textbf{Set XXXII}} : metl, {$H_{T}$}, topmassl, aplan, sqrts
191: \item \textit{\textbf{Set XXXIII}} : metl, spher, costhetastar, aplan, cent
192: % \item \textit{\textbf{Set XXXIV}} : metl, spher, sqrts, topmassl, ktminp
193: \end{itemize}
194:
195: The criteria used for making a decision on which variable should be used follow:
196: \begin{itemize}
197: \item No more than 5 variables to keep NN simple and stable. More variables leads to instabilities (different
198: result after each retraining) and require larger training samples.
199: % \item We want to use $metl$ (\met significance) variable, since it's the one providing best discrimination.
200: \item We do not want to use highly correlated variables in same NN.
201: % \item We can not use tau-based variables.
202: \item We want to use variables with high discriminating power.
203: \end{itemize}
204:
205: In order to make the decision about which of these 11 choices is the optimal we created an ensemble of
206: 20000 pseudo-datasets each containing events randomly (according to a Poisson distribution) picked
207: from QCD, EW and $\ttbar$ templates. Each of these datasets was treated like real data, meaning applying all
208: the cuts and doing the shape fit of event topological NN. QCD templates for fit were made from the same
209: ``loose-tight $\tau$ sample'' from which the QCD component of the ``data'' was drawn.
210: The figure of merit chosen is given by Equation \ref{merit} below:
211:
212: \begin{equation}
213: f = \displaystyle \frac{(N_{fit} - N_{true})}{N_{true}}
214: \label{merit}
215: \end{equation}
216:
217:
218: \noindent where $N_{fit}$ is the number of $t\bar{t}$ pairs given by
219: the fit and $N_{true}$ is the number of $t\bar{t}$ pairs from the Poisson distribution.
220: In both Set and $\not\!\! E_{T}$ significance optimization, the lowest RMS was used
221: to caracterize which configuration is the best in each case.
222:
223:
224: The plots showing results concerning the set optimizations are found in Appendix \ref{app:set_opt} and are summarized
225: in Table \ref{setopt_table} below, where each RMS and mean are shown.
226: For NN training is standard to choose the number of hidden nodes as being twice
227: the number the number of variables used for the training. The parenthesis after each set ID show the number of
228: hidden nodes in NN training.
229:
230: \begin{table}[htbp]
231: \begin{tabular}{|c|r|r|r|} \hline
232: Set of variables & \multicolumn{1}{c|}{RMS} & \multicolumn{1}{c|}{mean} \\ \hline
233:
234: \hline
235:
236:
237: Set1(6) & \multicolumn{1}{c|}{0.1642} & \multicolumn{1}{c|}{0.0265}\\ \hline
238:
239: Set2(6) & \multicolumn{1}{c|}{0.1840} & \multicolumn{1}{c|}{0.0054}\\ \hline
240:
241: Set3(6) & \multicolumn{1}{c|}{0.1923} & \multicolumn{1}{c|}{0.0060}\\ \hline
242:
243: Set4(6) & \multicolumn{1}{c|}{0.1978} & \multicolumn{1}{c|}{0.0175}\\ \hline
244:
245: Set5(6) & \multicolumn{1}{c|}{0.2385} & \multicolumn{1}{c|}{0.0022}\\ \hline
246:
247: Set6(8) & \multicolumn{1}{c|}{0.1687} & \multicolumn{1}{c|}{0.0115}\\ \hline
248:
249: Set7(8) & \multicolumn{1}{c|}{0.1667} & \multicolumn{1}{c|}{0.0134}\\ \hline
250:
251: Set8(10) & \multicolumn{1}{c|}{0.1668} & \multicolumn{1}{c|}{0.0162}\\ \hline
252:
253: Set9(10) & \multicolumn{1}{c|}{0.1721} & \multicolumn{1}{c|}{0.0102}\\ \hline
254:
255: Set10(10) & \multicolumn{1}{c|}{0.1722} & \multicolumn{1}{c|}{0.0210}\\ \hline
256:
257: Se11(10) & \multicolumn{1}{c|}{0.1716} & \multicolumn{1}{c|}{0.0180}\\ \hline
258:
259: Set12(8) & \multicolumn{1}{c|}{0.1662} & \multicolumn{1}{c|}{0.0039}\\ \hline
260:
261: Set13(8) & \multicolumn{1}{c|}{0.1819} & \multicolumn{1}{c|}{0.0018}\\ \hline
262:
263: Set14(8) & \multicolumn{1}{c|}{0.1879} & \multicolumn{1}{c|}{0.0019}\\ \hline
264:
265: Set15(8) & \multicolumn{1}{c|}{0.1884} & \multicolumn{1}{c|}{-0.0004}\\ \hline
266:
267: Set16(6) & \multicolumn{1}{c|}{0.1912} & \multicolumn{1}{c|}{0.0034}\\ \hline
268:
269: Set17(6) & \multicolumn{1}{c|}{0.1768} & \multicolumn{1}{c|}{0.0074}\\ \hline
270:
271: Set18(6) & \multicolumn{1}{c|}{0.2216} & \multicolumn{1}{c|}{-0.0030}\\ \hline
272:
273: Set19(6) & \multicolumn{1}{c|}{0.1921} & \multicolumn{1}{c|}{0.0015}\\ \hline
274:
275: Set20(10) & \multicolumn{1}{c|}{0.1620} & \multicolumn{1}{c|}{0.0262}\\ \hline
276:
277: Set21(10) & \multicolumn{1}{c|}{0.1753} & \multicolumn{1}{c|}{0.0010}\\ \hline
278:
279: Set22(10) & \multicolumn{1}{c|}{0.1646} & \multicolumn{1}{c|}{0.0086}\\ \hline
280:
281: Set23(10) & \multicolumn{1}{c|}{0.1683} & \multicolumn{1}{c|}{0.0132}\\ \hline
282:
283: Set24(10) & \multicolumn{1}{c|}{0.2053} & \multicolumn{1}{c|}{0.0122}\\ \hline
284:
285: Set25(10) & \multicolumn{1}{c|}{0.1906} & \multicolumn{1}{c|}{0.0038}\\ \hline
286:
287: Set26(10) & \multicolumn{1}{c|}{0.2130} & \multicolumn{1}{c|}{0.0028}\\ \hline
288:
289: Set27(10) & \multicolumn{1}{c|}{0.1859} & \multicolumn{1}{c|}{0.0004}\\ \hline
290:
291: Set28(6) & \multicolumn{1}{c|}{0.1910} & \multicolumn{1}{c|}{-0.0022}\\ \hline
292:
293: Set29(8) & \multicolumn{1}{c|}{0.1587} & \multicolumn{1}{c|}{0.0214}\\ \hline
294:
295: Set30(10) & \multicolumn{1}{c|}{0.1546} & \multicolumn{1}{c|}{0.0148}\\ \hline
296:
297: Set31(10) & \multicolumn{1}{c|}{0.1543} & \multicolumn{1}{c|}{0.0203}\\ \hline
298:
299: Set32(10) & \multicolumn{1}{c|}{0.1468} & \multicolumn{1}{c|}{0.0172}\\ \hline
300:
301: Set33(10) & \multicolumn{1}{c|}{0.2201} & \multicolumn{1}{c|}{0.0081}\\ \hline
302:
303: %Set34(10) & \multicolumn{1}{c|}{0.1955} & \multicolumn{1}{c|}{0.0184}\\ \hline
304: \end{tabular}
305: \caption{Results for set optimization part whit $\not\!\! E_{T}$ significance $>$ 4.0 applied to all sets.
306: The number in parenthesis refers to number of hidden nodes in each case.}
307: \label{setopt_table}
308: \end{table}
309:
310: From Table \ref{setopt_table} we see that Set I has the lowest RMS, thus we chose it
311: as the set to be used in $\not\!\! E_{T}$ significance optimization part, whose results are
312: shown in Appendix \ref{app:metl_opt} and then summarized in Table \ref{metlopt_table} below
313:
314: \begin{table}[htbp]
315: \begin{tabular}{|c|r|r|r|} \hline
316: Set of variables & $\not\!\! E_{T}$ significance cut & RMS & \multicolumn{1}{c|}{mean} \\ \hline
317:
318: \hline
319:
320:
321: %Set6(10) & \multicolumn{1}{c|}{1.0} & \multicolumn{1}{c|}{0.2611} \\ \hline
322:
323: %Set6(10) & \multicolumn{1}{c|}{1.5} & \multicolumn{1}{c|}{0.2320} \\ \hline
324:
325: %Set6(10) & \multicolumn{1}{c|}{2.0} & \multicolumn{1}{c|}{0.2102} \\ \hline
326:
327: %Set6(10) & \multicolumn{1}{c|}{2.5} & \multicolumn{1}{c|}{0.2021} \\ \hline
328:
329: Set32(10) & \multicolumn{1}{c|}{3.0} & \multicolumn{1}{c|}{0.1507} & \multicolumn{1}{c|}{0.0157}\\ \hline
330:
331: Set32(10) & \multicolumn{1}{c|}{3.5} & \multicolumn{1}{c|}{0.1559} & \multicolumn{1}{c|}{0.0189}\\ \hline
332:
333: Set32(10) & \multicolumn{1}{c|}{4.0} & \multicolumn{1}{c|}{0.1468} & \multicolumn{1}{c|}{0.0172}\\ \hline
334:
335: Set32(10) & \multicolumn{1}{c|}{4.5} & \multicolumn{1}{c|}{0.1511} & \multicolumn{1}{c|}{0.0153}\\ \hline
336:
337: Set32(10) & \multicolumn{1}{c|}{5.0} & \multicolumn{1}{c|}{0.1552} & \multicolumn{1}{c|}{0.0205}\\ \hline
338:
339: %Set6(10) & \multicolumn{1}{c|}{5.5} & \multicolumn{1}{c|}{0.4008} \\ \hline
340: \end{tabular}
341: \caption{Results for $\not\!\! E_{T}$ significance optimization part when varying the $\not\!\! E_{T}$ significance cut
342: The number in parenthesis refers to number of hidden nodes in each case.}
343: \label{metlopt_table}
344: \end{table}
345:
346:
347:
348: Combined results from Tables \ref{setopt_table} and \ref{metlopt_table} show that the best configuration found
349: was Set I with $\not\!\! E_{T}$ significance $\geq$ 4.0. Therefore, this was the
350: configuration used to perform the cross-section measurement.Figure \ref{fig:METsig_RMS} shows the variation of the RMS as function
351: of the $\not\!\! E_{T}$ significance we applied.
352:
353: \begin{figure}[b]
354: \includegraphics[scale=0.4]{plots/METsig-RMS.eps}
355: \caption{Plot of RMS as a function the $\not\!\! E_{T}$ significance applied}
356: \label{fig:METsig_RMS}
357: \end{figure}
358:
359:
360: \clearpage
361:
362:
363: In order to check the validity of our emsemble tests procedure, it is instructive to plot both the
364: distribution of the predicted number of $t\bar{t}$ and what is called ``pull'', defined in Equation
365: \ref{pull} below:
366:
367: \begin{equation}
368: p = \displaystyle \frac{(N_{fit}-N_{true})}{\sigma_{fit}}
369: \label{pull}
370: \end{equation}
371:
372: \noindent where $\sigma_{fit}$ is the error on the number of $t\bar{t}$ pairs given by the fit.
373:
374: Figures \ref{fig:gaus_ttbar} and \ref{fig:pull} show both beforementioned distributions.
375:
376: From Figure \ref{fig:gaus_ttbar} we see a good agreement between the number of $t\bar{t}$ pairs
377: initially set in the ensemble and the measured value. And Figure \ref{fig:pull} shows a nice gaussian
378: curve, that indicates a good behaviour of the fit uncertainties in the ensembles.
379:
380: \begin{figure}[t]
381: \includegraphics[scale=0.5]{plots/gaus_ttbar.eps}
382: \caption{Distribution of the output ``measurement'' for an ensemble with 116.9 $\ttbar$ events.}
383: \label{fig:gaus_ttbar}
384: \end{figure}
385:
386: \begin{figure}[t]
387: \includegraphics[scale=0.5]{plots/pull1-40.eps}
388: \caption{The ensemble test's pull.}
389: \label{fig:pull}
390: \end{figure}
391:
392:
393: %\newpage
394:
395:
396:
397:
398:
399: \clearpage
400:
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