
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 $f(x)=|x-2|-|x+1|$, $g(x)=-x$.  
$(1)$ Solve the inequality $f(x) > g(x)$;  
$(2)$ For any real number $x$, the inequality $f(x)-2x\leqslant 2g(x)+m$ ($m\in\mathbb{R}$) always holds, find the minimum value of the real number $m$.

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