
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
Consider the sequence $\{a_n\}$, which is an arithmetic sequence, and $a_{11} = \frac{\pi}{2}$. Let $f(x) = \sin 2x + 2\cos^2\frac{x}{2}$, and define $b_n = f(a_n)$. Find the sum of the first 21 terms of the sequence $\{b_n\}$.

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