
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 7-tuples $(b_0, b_1, \dots, b_6),$ where each entry is either 0 or 1, so $T$ contains $2^7$ 7-tuples. For each 7-tuple $t = (b_0, b_1, \dots, b_6)$ in $T,$ let $q_t(x)$ be the polynomial of degree at most 6 such that
\[q_t(n) = b_n\] for $0 \le n \le 6.$ Calculate
\[\sum_{t \in T} q_t(7).\]

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