
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 $g(x) = (x^2 - \cos x) \sin \frac{\pi}{6}$, for any $x\_1$ and $x\_2$ in the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, the following conditions are satisfied:

1. $x\_1^3 > x\_2^3$
2. $|x\_1| > x\_2$
3. $x\_1 > |x\_2|$
4. $x\_1^2 > x\_2^2$

Determine the condition(s) that always guarantee $g(x\_1) > g(x\_2)$.

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