
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 $f(x) = \sin\left(\frac{\pi}{2} - x\right)\sin x - \sqrt{3}\cos^2 x + \frac{\sqrt{3}}{2}$.
1. Find the smallest positive period of $f(x)$.
2. Discuss the monotonicity of $f(x)$ on the interval $\left[\frac{\pi}{6}, \frac{5\pi}{6}\right]$ and find the minimum value of $f(x)$ on this interval.

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