
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 functions $f(x)=|x-3|-5$ and $g(x)=|x+2|-2$.

$(1)$ Find the solution set for the inequality $f(x) \leqslant 2$;  
$(2)$ If the inequality $f(x)-g(x) \geqslant m-3$ has solutions, find the range of the real number $m$.

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