
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
In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, $a_1$, $a_2$, and $a_5$ form a geometric sequence, and the sum of the first $10$ terms of this sequence is $100$. The sum of the first $n$ terms of the sequence $\{b_n\}$ is $S_n$, and it satisfies $S_n=2b_n-1$.

$(I)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;

$(II)$ Let $C_n=a_n+\log_{\sqrt{2}} b_n$. The sum of the first $n$ terms of the sequence $\{C_n\}$ is $T_n$. If the sequence $\{d_n\}$ is an arithmetic sequence, and $d_n= \frac{T_n}{n+c}$, where $c\neq 0$.

$(i)$ Find the non-zero constant $C$;

$(ii)$ If $f(n)=\frac{d_n}{(n+36)d_{n+1}}$ $(n\in \mathbb{N}^*)$, find the maximum value of the term in the sequence $\{f(n)\}$.

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