
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]


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Let $\{a_n\}$ be a geometric sequence with a common ratio greater than $1$, and let $S_n$ be the sum of the first $n$ terms of the sequence. Given that $S_3 = 7$, and $a_1$, $a_2$, $a_3 - 1$ form an arithmetic sequence.

1. Find the general term formula for the sequence $\{a_n\}$.
2. If $b_n = \log_4 a_{2n+1}$, $n=1, 2, 3, \ldots$, find the sum: $\frac{1}{b_1b_2} + \frac{1}{b_2b_3} + \frac{1}{b_3b_4} + \ldots + \frac{1}{b_{n-1}b_n}$.

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