
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
In the Cartesian coordinate system, given rays $OA$: $x-y=0 (x\geqslant 0)$, $OB$: $2x+y=0 (x\geqslant 0)$. A line passing through point $P(1,0)$ intersects rays $OA$ and $OB$ at points $A$ and $B$ respectively.

(1) When the midpoint of $AB$ is on the line $x-2y=0$, find the equation of line $AB$;

(2) When the area of $\triangle AOB$ is minimized, find the equation of line $AB$;

(3) When $PA\cdot PB$ is minimized, find the equation of line $AB$.

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