
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
Let $\{a_n\}$ be an arithmetic sequence, with the sum of the first $n$ terms denoted as $S_n$. Let $\{b_n\}$ be a monotonically increasing geometric sequence, with $b_1=2$ being the arithmetic mean of $a_1$ and $a_2$, and $a_3=5$. If $b_3=a_4+1$, and for $n \geq m$, $S_n \leq b_n$ always holds, then the minimum value of $m$ is $\boxed{4}$.

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