
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 $x, y \in \mathbb{R}$, $m+n=7$, $f(x)=|x-1|-|x+1|$.  
$(1)$ Solve the inequality $f(x) \geqslant (m+n)x$;  
$(2)$ Let $max\{a,b\}= \begin{cases} a & (a \geqslant b) \\ b & (a < b) \end{cases}$, find the minimum value of $F=max\{|x^{2}-4y+m|,|y^{2}-2x+n|\}$.

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