
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
Find the maximum value of
\[
\begin{vmatrix}
1 & 1 & 1 \\
1 & 1 + \sin^2 \theta & 1 \\
1 + \cos^2 \theta & 1 & 1
\end{vmatrix},
\]
as $\theta$ ranges over all angles.

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