
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 $\triangle ABC$ with angles $A$, $B$, $C$ and corresponding side lengths $a$, $b$, $c$. Let $\vec{m} = (\cos A, \cos C)$, $\vec{n} = (c, a)$, $\vec{p} = (2b, 0)$, and $\vec{m} \cdot (\vec{n} - \vec{p}) = 0$.

1. Find the measure of angle $A$.
2. Find the range of the function $f(x) = \sin x \cos x + \sin x \sin (x - \frac{\pi}{6})$ when $|x| \leq A$.

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