
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 geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, satisfying $S_4=2a_4-1$ and $S_3=2a_3-1$.
$(1)$ Find the general formula for $\{a_n\}$;
$(2)$ Let $b_n=\log_{\sqrt{2}}\left(\frac{16}{S_n+1}\right)$, find the maximum value of $b_1+b_2+\ldots+b_n$.

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