
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 function $f(x)=x^{2}-2\cos x$, for any $x_{1}$ and $x_{2}$ within the interval $\left[-\frac{2\pi}{3}, \frac{2\pi}{3}\right]$, the following conditions are satisfied:
1) $x_{1} > x_{2}$;
2) $x_{1}^{2} > x_{2}^{2}$;
3) $x_{1} > |x_{2}|$;
4) $|x_{1}| > x_{2}$;

Determine the condition(s) that consistently guarantee $f(x_{1}) > f(x_{2})$. (Choose the corresponding number(s))

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