
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 that the sum of the first $n$ terms of an increasing arithmetic sequence ${a_n}$ is $S_n$, $a_4a_6=117$, and $a_3+a_7=22$.

1. Find $S_n$;
2. Find the sum of the first $n$ terms of the sequence $\{\frac{1}{S_n+3n+6}\}$, denoted as $T_n$.

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