
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 triangle \( \triangle ABC \), the sides opposite angles \( A \), \( B \), and \( C \) are denoted as \( a \), \( b \), and \( c \) respectively, and it satisfies \( 2c\cos B = 2a + b \).
\( (1) \) Find angle \( C \);
\( (2) \) If the area of \( \triangle ABC \) is \( S = \frac { \sqrt {3}}{2}c \), find the minimum value of \( ab \).

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