
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
Evaluate the following determinant:
\[
\begin{vmatrix} 
\cos \alpha \cos \beta & \cos \alpha \sin \beta & -\sin (\alpha + \gamma) \\ 
-\sin \beta & \cos \beta \cos \gamma & \sin \beta \sin \gamma \\ 
\sin (\alpha + \gamma) \cos \beta & \sin (\alpha + \gamma) \sin \beta & \cos (\alpha + \gamma)
\end{vmatrix}.
\]

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