
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 $ABC$, with sides $a, b, c$ opposite to angles $A, B, C$ respectively, it is known that $2\sin^2 A + \sin^2 B = \sin^2 C$.
(1) If $b = 2a = 4$, find the area of $\triangle ABC$;
(2) Find the minimum value of $\frac{c^2}{ab}$ and determine the value of $\frac{c}{a}$ at this minimum.

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