
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 8-tuples \( (b_0, b_1, \dots, b_7) \), where each entry is 0 or 1, so \( T \) contains \( 2^8 \) 8-tuples. For each 8-tuple \( t = (b_0, b_1, \dots, b_7) \) in \( T \), let \( q_t(x) \) be the polynomial of degree at most 7 such that
\[ q_t(n) = b_n \] for \( 0 \le n \le 7 \).
Find
\[ \sum_{t \in T} q_t(8). \]

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