
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 $\triangle ABC$, $\angle C = 2 \angle A$, $\cos \angle A = \frac{3}{4}$, and $\overrightarrow{BA} \cdot \overrightarrow{BC} = \frac{27}{2}$. Find
(1) the value of $\cos \angle B$;
(2) the length of side $AC$.

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