
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
Let  $f(x)=(x + a)(x + b)$  where  $a,b>0$ . For any reals  $x_1,x_2,\ldots ,x_n\geqslant 0$  satisfying  $x_1+x_2+\ldots +x_n =1$ , find the maximum of  $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $ .

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