
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
Let $A$ and $B$ be the angles of a triangle with $C = 90^\circ - A$. Compute
\[
\begin{vmatrix} \cot A & 1 & \frac{1}{2} \\ \frac{1}{2} & \cot B & 1 \\ 1 & \frac{1}{2} & \cot C \end{vmatrix}.
\]

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