
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 $(b_0, b_1, \dots, b_{11})$, where each $b_i$ is either 0 or 1. Thus, $T$ consists of $2^{12}$ 12-tuples. Define $q_t(x)$ for each 12-tuple $t = (b_0, b_1, \dots, b_{11}) \in T$ as a polynomial of degree at most 11 such that
\[q_t(n) = b_n\] for every integer $n$ from 0 to 11.
Find the sum:
\[\sum_{t \in T} q_t(12).\]

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