
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 $x, y \in \mathbb{R}$, $m + n = 7$, and $f(x) = |x - 1| - |x + 1|$.

(1) Solve the inequality $f(x) \geq (m + n)x$;

(2) Let $\max\{a, b\} = \begin{cases} a, & (a \geq b) \\ b, & (a < b) \end{cases}$, 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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