
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, so $S$ contains $2^{11}$ 11-tuples. For each 11-tuple $s = (a_0, a_1, \dots, a_{10})$ in $S$, let $p_s(x)$ be the polynomial of degree at most 10 such that
\[p_s(n) = a_n\] for $0 \le n \le 10.$ For example, $p(x) = p_{(0,1,0,1,0,1,0,1,0,1,0)}(x)$ is the polynomial of degree at most 10 such that $p(0) = p(2) = p(4) = p(6) = p(8) = p(10) = 0$ and $p(1) = p(3) = p(5) = p(7) = p(9) = 1.$

Find
\[\sum_{s \in S} p_s(11).\]

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