
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 $\triangle ABC$, let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $\cos 2C - \cos 2A = 2\sin\left( \frac{\pi}{3} + C\right) \sin\left( \frac{\pi}{3} - C\right)$.

(1) Find the value of angle $A$.

(2) If $a = \sqrt{3}$ and $b \geq a$, find the range of $2b - c$.

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