
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 points of the lines $2x-y-3=0$ and $4x-3y-5=0$, and is perpendicular to the line $x+y-2=0$.  
$(1)$ Find the equation of the line $l$.  
$(2)$ If a circle $C$ passes through the point $(1,0)$ and its center lies on the positive $x$-axis, and the chord cut off by the line $l$ on the circle $C$ has a length of $2\sqrt{2}$, find the standard equation of the circle $C$.

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