
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 $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\left(2b-c\right)\cos A=a\cos C$.<br/>$(1)$ Find the value of $A$;<br/>$(2)$ If the shortest distance between the axis of symmetry of the graphs of the functions $y=\cos \left(\omega x+A\right)$ and $y=\sin(\omega x+\frac{π}{6})(\omega>0)$ is $\frac{π}{6}$, find the value of $\omega$.

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