
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
Set the origin as the pole and the non-negative half-axis of the $x$-axis as the polar axis to establish a polar coordinate system. Given $A(2,\pi)$, $B(2, \dfrac {\pi}{2})$, the polar equation of circle $C$ is $\rho^{2}-6\rho\cos \theta+8\rho\sin \theta+21=0$. $F$ is any point on circle $C$.  
$(1)$ Write the parametric equation of circle $C$;  
$(2)$ Find the maximum value of the area of $\triangle ABF$.

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