
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 $f(x)=|x-5|-|x-2|$.  
$(1)$ If there exists $x \in \mathbb{R}$, such that $f(x) \leqslant m$ holds, find the range of $m$;  
$(2)$ Find the solution set of the inequality $x^{2}-8x+15+f(x) \leqslant 0$.

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