
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 $S\_n$, vector $\overrightarrow{a} = (S\_n, n)$, $\overrightarrow{b} = (9n - 7, 2)$, and $\overrightarrow{a}$ is collinear with $\overrightarrow{b}$.

1. Find the general formula for the sequence $\{a\_n\}$.
2. For any $m ∈ ℕ^∗$, let $b\_m$ denote the number of terms in the sequence $\{a\_n\}$ that fall within the interval $(9^m, 9^{2m})$. Find the sum of the first $m$ terms of the sequence $\{b\_m\}$, denoted as $S\_m$.

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