
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 the function $f(x)=a\sin \omega x-\cos \omega x$ where $(a \gt 0, \omega \gt 0)$ has a maximum value of $2$, find $a=\_\_\_\_\_\_$. If a line $x=\frac{π}{m}$ is a line of symmetry for the graph of the function $f(x)$, where $m\in N^{*}$, then when $\omega$ takes the smallest integer value, the function $f(x)$ achieves its maximum value $______$ times within the interval $(0,10)$.

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