
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 the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1 (a > b > 0)$, with the foci $F_1$ and $F_2$, let $P$ be a point on the ellipse where the magnitude of the projection of $\overrightarrow{F_1 F_2}$ onto $\overrightarrow{F_1 P}$ is exactly $|\overrightarrow{F_1 P}|$, and the angle between them is $\dfrac{\pi}{6}$. Calculate the eccentricity $e$ of the ellipse.

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