
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 $2^{12}$ 12-tuples. For each 12-tuple $t = (a_0, a_1, \dots, a_{11})$ in $T$, let $q_t(x)$ be the polynomial of degree at most 11 such that
\[q_t(n) = a_n\] for $0 \le n \le 11.$ 
Find
\[\sum_{t \in T} q_t(12).\]

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