
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 point $P(2,1)$ and point $Q$ is symmetric to $P$ with respect to the origin $O$. The lines $PM$ and $QM$ intersect at point $M$, and the product of their slopes is $-\dfrac{1}{4}$.

$(1)$ Find the equation of the trajectory $C$ of point $M$;

$(2)$ A line $l$ passes through $P$ and intersects the trajectory $C$ at another point $A$. Find the range of the area of $\triangle PAO$.

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