
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 the line $l$ passes through the intersection point of the lines $l_{1}$: $2x+y-5=0$ and $l_{2}$: $x-2y=0$.

$(1)$ If the line $l$ is parallel to the line $x-y=0$, find the equation of the line $l$.

$(2)$ If the area of the triangle formed by the line $l$ and the positive half-axes of the coordinate system is $4$, find the equation of the line $l$.

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