
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 $S\_n$, where $a\_1=1$ and $3S\_n = a_{n+1} - 1$.

1. Find the general formula for the sequence $\{a\_n\}$.
2. Consider an arithmetic sequence $\{b\_n\}$ with its first $n$ terms sum $T\_n$, where $a\_2 = b\_2$ and $T\_4 = 1 + S\_3$. Find the value of $\frac{1}{b\_1 \cdot b\_2} + \frac{1}{b\_2 \cdot b\_3} + \dots + \frac{1}{b_{10}b_{11}}$.

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