
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A sequence  $y_1, y_2, \ldots, y_k$  of real numbers is called  $\textit{zigzag}$  if  $k=1$ , or if  $y_2-y_1, y_3-y_2, \ldots, y_k-y_{k-1}$  are nonzero and alternate in sign. Let  $X_1, X_2, \ldots, X_n$  be chosen independently from the uniform distribution on  $[0,1]$ . Let  $a\left(X_1, X_2, \ldots, X_n\right)$  be the largest value of  $k$  for which there exists an increasing sequence of integers  $i_1, i_2, \ldots, i_k$  such that  $X_{i_1}, X_{i_2}, \ldots X_{i_k}$  is zigzag. Find the expected value of  $a\left(X_1, X_2, \ldots, X_n\right)$  for  $n \geq 2$ .

Present the answer in LaTex format: \boxed{Your answer}
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