
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 $\triangle ABC$, the sides opposite to angles A, B, C are $a$, $b$, $c$ respectively, with $a=\frac{3\sqrt{6}}{2}$, $A=60^\circ$, and $C=45^\circ$.
(1) Find the value of $c$;
(2) Construct a new $\triangle ABP$ outside of $\triangle ABC$ (with $C$ not on the same side of $AB$ as $P$) such that $\angle APB = 30^\circ$, and find the maximum area of $\triangle ABP$.

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