
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 $T$ be the set of 12-tuples $(a_0, a_1, \dots, a_{11})$, where each entry is 0 or 1. Thus, $T$ contains $2^{12}$ such 12-tuples. Define a polynomial $q_s(x)$ of degree at most 11 for each 12-tuple $s = (a_0, a_1, \dots, a_{11})$ in $T$ such that
\[ q_s(n) = a_n \] for $0 \le n \le 11$.
Determine
\[ \sum_{s \in T} q_s(12). \]

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