
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
Given a triangle ABC with internal angles A, B, C and their corresponding opposite sides a, b, c, such that $c^2 = ab$, and $3b \cos C = 2(a \cos C + c \cos A)$. When $a + b = \_\_\_\_\_\_$, the area of the triangle ABC equals $2 \sqrt {5}$.

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