
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
In the triangle \( \triangle ABC \), the lengths of the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( a \), \( b \), and \( c \) respectively. Given that:
$$
\begin{array}{l}
\angle C = \max \{\angle A, \angle B, \angle C\}, \\
\sin C = 1 + \cos C \cdot \cos (A-B), \\
\frac{2}{a} + \frac{1}{b} = 1 .
\end{array}
$$

Find the minimum perimeter of the triangle \( \triangle ABC \).

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