
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 two fixed points $A(-2,0)$ and $B(-1,0)$, if a moving point $P$ on curve $C$ satisfies $|PA|=\sqrt{2}|PB|$, and point $Q(x_{0},y_{0})$ is a moving point on line $l$: $x+y-4=0$, $QE$ and $QF$ are two tangents to curve $C$ with $E$ and $F$ being the points of tangency. When the area of quadrilateral $OEQF$ (with point $O$ being the origin) is minimized, the equation of line $EF$ is _______.

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