
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 a sequence $\{ a\_n\}$ with its first $n$ terms sum $S\_n$, it satisfies the equation $a\_n + 2S\_n \cdot S_{n-1} = 0 (n \geqslant 2)$. If $S\_5 = \frac{1}{11}$, then $a\_1 = (\quad \quad)$.

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