
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  $x$  be in the interval  $\left(0, \frac{\pi}{2}\right)$  such that  $\sin x - \cos x = \frac12$  . Then  $\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}$  , where  $m, n$ , and  $p$  are relatively prime positive integers, and  $p$  is not divisible by the square of any prime. Find  $m + n + p$ .

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