
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 equation of an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, where $A$ and $B$ are the left and right vertices, and $F$ is the right focus. If the maximum distance from any point on the ellipse to focus $F$ is 3, and the eccentricity is a root of the equation $2x^{2}-5x+2=0$,

1. Find the standard equation of the ellipse;
2. If $P$ is any point on the ellipse, and $AP$ and $PB$ are extended to intersect line $l$: $x=4$ at points $M$ and $N$, find the minimum value of line segment $MN$.

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