
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
Given points $A(-1,0)$, $B(1,0)$, and a moving point $M$ whose trajectory curve $C$ satisfies $\angle AMB = 2\theta$ and $|\overrightarrow{AM}| \cdot |\overrightarrow{BM}| \cos^2\theta = 3$. A line passing through point $B$ intersects curve $C$ at points $P$ and $Q$.  
$(1)$ Find the value of $|\overrightarrow{AM}| + |\overrightarrow{BM}|$ and write the equation of curve $C$;  
$(2)$ Find the maximum area of $\triangle APQ$.

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