
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 13-tuples $(a_0, a_1, \dots, a_{12}),$ where each entry is either 0, 1, or 2, such that each $a_i$ equals 2 only once among all tuples, with all other entries being 0 or 1. Thus, $T$ contains $12 \cdot 2^{12}$ 13-tuples. For each 13-tuple $t = (a_0, a_1, \dots, a_{12})$ in $T$, let $q_t(x)$ be the polynomial of degree at most 12 such that
\[q_t(n) = a_n\] for $0 \le n \le 12$. Find
\[\sum_{t \in T} q_t(14).\]

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