
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
Compute the determinant of the following matrix:
\[
\begin{vmatrix} \cos \frac{\pi}{4} & \cos \frac{\pi}{2} & \cos \frac{3\pi}{4} \\ \cos \pi & \cos \frac{5\pi}{4} & \cos \frac{3\pi}{2} \\ \cos \frac{7\pi}{4} & \cos 2\pi & \cos \frac{9\pi}{4} \end{vmatrix}
\]
All the angles are in radians.

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