
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
Let $P$ be a point on the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\ (a > b > 0)$, and let $F_{1}$, $F_{2}$ be the foci. If $\angle PF_{1}F_{2}=75^{\circ}$ and $\angle PF_{2}F_{1}=15^{\circ}$, determine the eccentricity of the ellipse.

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