
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 $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{4}=1$ $(a > 2)$, and a point $P$ on it, the sum of the distances from $P$ to its two foci $F_1$ (left) and $F_2$ (right) is $6$,

$(1)$ Find the eccentricity of the ellipse $C$.

$(2)$ If $PF_2 \perp x$-axis, and the projection of $P$ on the $y$-axis is point $Q$, find the coordinates of point $Q$.

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