
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 $\{a_{n}\}$ that satisfies: $a_{1}=3$, $a_{2}=6$, $a_{3}=11$, starting from the second term, each term minus the previous term forms an arithmetic sequence.<br/>$(1)$ Find $a_{n}$;<br/>$(2)$ Let $b_n=\frac{a_n}{3^{\sqrt{a_n-2}}}$, if $b_{n}\leqslant m$ always holds, find the range of values for $m$.

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