
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
It is known that $\{b_n\}$ is a geometric sequence with a common ratio greater than $1$, and $b_1$, $b_3$ are the two roots of the function $f(x)=x^2-5x+4$.

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

$(2)$ If the sequence $\{a_n\}$ satisfies $a_n=\log_2b_n+n+2$, and $a_1+a_2+a_3+...+a_m \leqslant 63$, find the maximum value of $m$.

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