
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 an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, $S_5=35$, and the arithmetic mean of $a_5$ and $a_7$ is $13$.

$(1)$ Find $a_n$ and $S_n$;

$(2)$ Let $b_n = \dfrac{4}{a_n^2-1}$ $(n \in \mathbb{N}^*)$, calculate the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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