
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 an ellipse $C$ with its center at the origin, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively, the major axis is $6$, and the eccentricity is $\dfrac{\sqrt{5}}{3}$  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Given a point $P$ on the ellipse $C$, and $PF_{1}=4$, find the distance from point $P$ to the right directrix.

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