
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 \( C: (x-2)^{2}+\frac{y^{2}}{4}=1 \), the line \( l: x-y+1=0 \), and the point \( R(1,4) \):
1. Find a point \( M \) inside the ellipse \( C \) such that for any line passing through \( M \) intersecting line \( l \) at point \( Q \) and intersecting ellipse \( C \) at points \( A \) and \( B \), the condition \( \frac{|A M|}{|M B|}=\frac{|A Q|}{|Q B|} \) holds. Does such a point \( M \) exist?
2. Draw a line through the point \( R \) intersecting the ellipse at points \( E \) and \( F \). If \( |R E|=|E F| \), find the equation of the line \( E F \).

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