
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 $x^{2}+4y^{2}=4$, touching the ellipse at points $A$ and $B$. When point $P$ moves, determine the coordinates of the fixed point through which the line $AB$ passes.

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