
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 $f(x)=\sin x-m\cos x$ achieves its maximum value at $x=\frac{π}{3}$.  
$(1)$ Find $m$;  
$(2)$ In triangle $ABC$, where the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $f(A-\frac{π}{2})=0$, $2b+c=3$, find the minimum value of $a$.

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