
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 $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. Let $f(x) = a^2x^2 - (a^2 - b^2)x - 4c^2$.
1. If $f(1) = 0$ and $B - C = \frac{\pi}{3}$, find the measure of angle $C$.
2. If $f(2) = 0$, find the range of values for angle $C$.

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