
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 $\triangle ABC$, where $A$, $B$, and $C$ are the three interior angles and $a$, $b$, and $c$ are the sides opposite these angles respectively, it is known that $b^2$, $a^2$, and $c^2$ form an arithmetic progression.

(1) Find the minimum value of $\cos A$;
(2) If $a = 2$, what is the maximum area of triangle $\triangle ABC$ when angle $A$ is at its largest?

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