
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  $a$ ,  $b$ ,  $c$  be the sides of an acute triangle  $\triangle ABC$  , then for any  $x, y, z \geq 0$ , such that  $xy+yz+zx=1$  holds inequality:  $$ a^2x + b^2y + c^2z \geq 4F $$   where  $F$  is the area of the triangle  $\triangle ABC$ 

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