
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 that $\{a_n\}$ is a geometric sequence and $\{b_n\}$ is an arithmetic sequence. If $a_1 \cdot a_6 \cdot a_{11} = -3\sqrt{3}$ and $b_1 + b_6 + b_{11} = 7\pi$, find the value of $\tan \left( \frac{b_3 + b_9}{1 - a_4 \cdot a_8} \right)$.
Options:
A) 1
B) $\frac{\sqrt{2}}{2}$
C) $-\frac{\sqrt{2}}{2}$
D) $-\sqrt{3}$

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