
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 $P(a,b)$ be a moving point on the ellipse $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, and satisfies $|P{F}_{2}|=|{F}_{1}{F}_{2}|$.

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

(2) Let line $PF_{2}$ intersect the ellipse at points $A$ and $B$. If line $PF_{2}$ intersects the circle $(x+1{)}^{2}+(y- \sqrt{3}{)}^{2}=16$ at points $M$ and $N$, and $|MN|= \frac{5}{8}|AB|$, find the equation of the ellipse.

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