
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 acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $\frac{{bcosC}}{{ccosB}}=\frac{{1+cos2C}}{{1+cos2B}}$.<br/>$(1)$ If $cosA=\frac{1}{3}$, find $\cos B$;<br/>$(2)$ If $b=5$ and $cosA=\frac{3}{4}$, find $a$.

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