
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and it is known that $a_1=1$, $S_n+1=a_{n+1}$ $(n\in\mathbb{N}^*)$. The sequence $\{b_n\}$ is an arithmetic sequence with the first term being a positive number, and the sum of the first $n$ terms of the sequence $\left\{\frac{1}{b_nb_{n+1}}\right\}$ is $\frac{n}{n+1}$.

$(1)$ Find the general formulas for $a_n$ and $b_n$ of the sequences $\{a_n\}$ and $\{b_n\}$;

$(2)$ Let $c_n= \frac{b_n}{4a_n}$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ is $T_n$. Find the range of values of $m$ such that the inequality $m\leqslant 2^n(1-T_n)$ holds for any positive integer $n$.

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