
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 an acute triangle $ABC$, points $R$ and $S$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $RS$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $Z$ and $W$. Suppose $ZR=12$, $RS=30$, and $SW=18$. Find $AB\cdot AC$ in the form $p\sqrt q$, where $p$ and $q$ are positive integers, and $q$ is not divisible by the square of any prime.

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