
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 that point \( P \) is a moving point on the line \( x + 2y = 4 \), and through point \( P \) two tangents are drawn to the ellipse \( x^{2} + 4y^{2} = 4 \). The points of tangency are \( A \) and \( B \). When point \( P \) moves, the line \( AB \) passes through a fixed point. Determine the coordinates of this fixed point.

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