
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 $S_n$ be the sum of the first n terms of an arithmetic sequence $\{a_n\}$, and $T_n$ be the product of the first n terms of a geometric sequence $\{b_n\}$. Given that the common difference $d$ of the arithmetic sequence $\{a_n\}$ is not equal to $0$, and for every positive integer $n$ less than $2011$, $S_n = S_{2011-n}$ holds, it can be derived that $a_{1006} = 0$. Given that the common ratio $q$ of the geometric sequence $\{b_n\}$ is not equal to $1$, and for every positive integer $n$ less than $23$, $T_n = T_{23-n}$ holds, then find the value of $b_{11}$.

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