
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
There exist constants $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ such that
\[
\cos^7 \theta = a_1 \cos \theta + a_2 \cos 2 \theta + a_3 \cos 3 \theta + a_4 \cos 4 \theta + a_5 \cos 5 \theta + a_6 \cos 6 \theta + a_7 \cos 7 \theta
\]
for all angles $\theta.$ Find $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2 + a_7^2.$

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