
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
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$, given that $a_1 + a_{13} = 26$ and $S_9 = 81$.

$(1)$ Find the general formula for $\{a_n\}$.

$(2)$ If $b_n = \frac{1}{a_{n+1}a_{n+2}}$, and $T_n = b_1 + b_2 + \ldots + b_n$, and $30T_n - m \leqslant 0$ holds for all $n \in \mathbb{N}^*$, find the minimum value of the real number $m$.

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