
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 the functions $f(x) = \frac{1}{2}(x + \frac{1}{x})$, $g(x) = \frac{1}{2}(x - \frac{1}{x})$.

(1) Find the zeros of the function $h(x) = f(x) + 2g(x)$;

(2) Find the minimum value of the function $F(x) = [f(x)]^{2n} - [g(x)]^{2n}$ ($n \in \mathbb{N}^*$).

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