
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 $\triangle ABC$, it is given that $\overrightarrow{AB} \cdot \overrightarrow{AC} = 9$, $\sin B = \cos A \cdot \sin C$, $S_{\triangle ABC} = 6$, and $P$ is a point on line segment $AB$ such that $\overrightarrow{CP} = x \cdot \frac{\overrightarrow{CA}}{|\overrightarrow{CA}|} + y \cdot \frac{\overrightarrow{CB}}{|\overrightarrow{CB}|}$. Find the minimum value of $\frac{1}{x} + \frac{1}{y}$.

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