
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, respectively, and F is the right focus. If the maximum distance from a point on the ellipse to the 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 point P is any point on the ellipse, extend lines AP and PB to intersect line $l: x=4$ at points M and N, respectively. Find the minimum value of segment MN.

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