
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 that $max\\{a,b\\}= \\begin{cases} a & (a\\geqslant b) \\\\ b & (a < b)\\end{cases}$, where $x$, $y∈R$, and $m+n=6$, find the minimum value of $F=max\\{|x^{2}-4y+m|,|y^{2}-2x+n|\\}$.

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