
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 \(\frac{y}{4} + x^{2} = 1\), let \(P\) be an arbitrary point on the ellipse. Draw lines through point \(P\) that are parallel to \(l_1: y=2x\) and \(l_2: y=-2x\). These lines intersect the lines \(l_2\) and \(l_1\) at points \(M\) and \(N\) respectively. Find the maximum value of \(|MN|\).

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