
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
Expand the x-coordinate of each point on the graph of the function $g(x) = \sin(\omega x - \phi)$ ($\omega > 0$, $0 < \phi < \pi$) by a factor of 2 (keeping the y-coordinate unchanged) and then translate it left by $\frac{\pi}{6}$ unit length to obtain the graph of the function $y = f(x)$. Suppose the graph of $f(x)$ passes through the point $\left(\frac{\pi}{6}, 0\right)$, and the distance between two adjacent axes of symmetry is $\frac{\pi}{2}$.

1. Find the values of $\omega$ and $\phi$.
2. Find the intervals of increase for $y = f(x)$.
3. If $\frac{\pi}{6} < A < \frac{\pi}{3}$, find the range of values for $f(A)$.

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