
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 $\left\{a_n\right\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, with $S_{11}= \frac{11}{3}\pi$. Let $\left\{b_n\right\}$ be a geometric sequence, and $b_4, b_8$ be the two roots of the equation $4x^2+100x+{\pi}^2=0$. Find the value of $\sin \left(a_6+b_6\right)$.

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