
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 $\{a\_n\}$ where the sum of its first $n$ terms, $S\_n$, satisfies the equation $$\frac{1}{\sqrt{S\_{n-1}}} - \frac{1}{\sqrt{S\_n}} - \frac{1}{\sqrt{S\_n S\_{n-1}}} = 0$$, and $a\_1 = 1$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Within the first 100 terms of the sequence $\{a\_n\}$, determine if there exist two terms $a\_m$ and $a\_t$ ($m, t \in \mathbb{N}^+$, and $m < t$) such that $\frac{1}{a\_2}$, $\frac{1}{a\_m}$, and $\frac{1}{a\_t}$ form a geometric sequence. If such terms exist, find all possible values of $m$ and $t$; otherwise, explain why.

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