
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 \(\frac{x^{2}}{6} + \frac{y^{2}}{2} = 1\) with the right focus \(F\). The line passing through \(F\) given by \(y = k(x-2)\) intersects the ellipse at points \(P\) and \(Q\) \((k \neq 0)\). If the midpoint of \(PQ\) is \(N\) and \(O\) is the origin, the line \(ON\) intersects the line \(x = 3\) at \(M\).

1. Find the angle \(\angle MFQ\).
2. Find the maximum value of \(\frac{|PQ|}{|MF|}\).

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