
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 $\{a_n\}$ be an arithmetic sequence, $S_n$ be the sum of its first $n$ terms, and given that $S_{11}= \frac{22\pi}{3}$, $\{b_n\}$ be a geometric sequence, and $b_5 \cdot b_7= \frac{\pi^2}{4}$, calculate the value of $\tan (a_6+b_6)$.

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