
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 entry is 0 or 1, so $T$ contains $2^{12}$ 12-tuples. For each 12-tuple $t = (b_0, b_1, \dots, b_{11})$ in $T$, let $q_t(x)$ be the polynomial of degree at most 11 such that
\[ q_t(n) = b_n \]
for $0 \le n \le 11.$ For example, $q(x) = q_{(0,1,0,0,1,0,1,0,0,0,1,0)}(x)$ is the polynomial of degree at most 11 such that $q(0) = q(2) = q(3) = q(5) = q(7) = q(8) = q(9) = q(11) = 0$ and $q(1) = q(4) = q(6) = q(10) = 1.$

Find
\[\sum_{t \in T} q_t(12).\]

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