
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 $\frac{x^{2}}{4}+ \frac{y^{2}}{2}=1$.
(1) M is a moving point on the line $l: \frac{x}{4}+ \frac{y}{2}=1$, and N is a moving point on the ellipse. Find the minimum value of |MN|.
(2) A chord AB is drawn through the point $P( \sqrt {2}, \frac{1}{2})$ on the ellipse such that $\overrightarrow {AP}=3 \overrightarrow {PB}$. Find the equation of the line containing the chord AB.

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