
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 the function $f(x) = \cos \omega x \cdot \sin(\omega x - \frac{\pi}{3}) + \sqrt{3} \cos^{2} \omega x - \frac{\sqrt{3}}{4} (\omega > 0, x \in R)$, and the distance from one of the symmetric centers of the graph of the function $y=f(x)$ to its axis of symmetry is $\frac{\pi}{4}$.
1. Find the value of $\omega$ and the equation of the axis of symmetry;
2. In $\triangle ABC$, the sides opposite to angles $A, B, C$ are respectively $a, b, c$. If $f(A)=0, \sin B=\frac{4}{5}, a=\sqrt{3}$, find the value of $b$.

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