
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 $f(x)=\sqrt{3}\sin{2x}+2\cos^2{x}+m$ has a maximum value of $1$ on the interval $[0,\frac{π}{4}]$.  
$(1)$ Find the value of the constant $m$;  
$(2)$ When $x\in \mathbb{R}$, find the minimum value of the function $f(x)$ and the corresponding set of $x$.

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