
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
Given a sequence $\left\{a_n\right\}$ satisfying $a_1=1, a_2=2, a_{n+2}=\left(1+\cos^2 \frac{n\pi}{2}\right)a_n+\sin^2 \frac{n\pi}{2}$, calculate the sum of the first 12 terms of this sequence.

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