
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 $\{a_n\}$ where each term is positive, and it satisfies $a_1=3$, $\frac{a_{n+1}+a_n}{n+1}=\frac{8}{a_{n+1}-a_n}$ ($n\in\mathbb{N}^*$).

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

$(2)$ If $b_n=\frac{1}{a_n^2-1}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $S_n$;

$(3)$ If the sequence $\{c_n\}$ satisfies $c_n=3^{\frac{a_n-1}{2}}+(-1)^{n-1}\lambda (a_{2^{n-1}}-1)$ ($\lambda$ is a non-zero constant), determine the range of values for $\lambda$ such that $c_{n+1} > c_n$ for all $n\in\mathbb{N}^*$.

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