
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 that $F$ is the right focus of the hyperbola $C$: ${{x}^{2}}-\dfrac{{{y}^{2}}}{3}=1$, and $P$ is a point on $C$ such that $PF$ is perpendicular to the $x$-axis. The coordinates of point $A$ are $(0,3)$. Determine the area of $\triangle APF$.

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