
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
Let $S$ be the set of 11-tuples $(a_0, a_1, \dots, a_{10})$, where each entry is either 0 or 1. The set $S$ contains $2^{11}$ such 11-tuples. For each 11-tuple $s = (a_0, a_1, \dots, a_{10})$ in $S$, define the polynomial $p_s(x)$ with degree at most 10 such that $p_s(n) = a_n$ for $0 \le n \le 10$. Find the sum:
\[
\sum_{s \in S} p_s(11).
\]

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