
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 $S$ be the set of 6-tuples $(a_0, a_1, \dots, a_5),$ where each entry is 0 or 1, so $S$ contains $2^6$ elements.  Each tuple $s = (a_0, a_1, \dots, a_5)$ in $S$ corresponds to a polynomial $p_s(x)$ of degree at most 5 such that
\[p_s(n) = a_n\]for $0 \le n \le 5.$  Determine the sum of all $p_s(12)$ for each $s \in S.$

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