
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 $\dfrac{x^{2}}{4}+ \dfrac{y^{2}}{2}=1$ with a point $P$ on it, and $F_{1}$, $F_{2}$ are the left and right foci of the ellipse, respectively, determine the number of points $P$ such that $\triangle F_{1}PF_{2}$ is a right-angled triangle.

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