
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, the rays OA and OB are given by $OA: x-y=0$ ($x\geq0$) and $OB: 2x+y=0$ ($x\geq0$), respectively. A line passing through point P(1, 0) intersects rays OA and OB at points C and D.
1. Find the equation of line CD when the area of triangle COP is equal to the area of triangle DOP.
2. Find the equation of line CD when the midpoint of CD lies on the line $x-2y=0$.

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