
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

If \( 0^{\circ} \leq \theta < 360^{\circ} \), the equation in \( \theta \): \( 3 \cos \theta + \frac{1}{\cos \theta} = 4 \) has \( p \) roots. Find \( p \).


If \( x - \frac{1}{x} = p \) and \( x^{3} - \frac{1}{x^{3}} = q \), find \( q \).


A circle is inscribed in an equilateral triangle of perimeter \( q \) cm. If the area of the circle is \( k \pi \) cm\(^{2} \), find \( k \).


Each interior angle of a regular polygon of \( k \) sides is \( m^{\circ} \). Find \( m \).

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