
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 the sequence $\{a_n\}$ be an arithmetic sequence with the first term $2$ and common difference $3$, and let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$, where $S_n=n^2-2n$  
$(1)$ Find the general formulas for $a_n$ and $b_n$ of the sequences $\{a_n\}$ and $\{b_n\}$ respectively.  
$(2)$ If the sum of the first $n$ terms of the sequence $\{a_n\}$ is $T_n$, find the maximum value of $n$ such that $T_n < 20b_n$.

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