
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 the following conditions for a line, find the equation of the line:<br/>
$(1)$ It passes through the intersection of the lines $x+2y-5=0$ and $3x-y-1=0$, and is parallel to the line $5x-y+100=0$;<br/>
$(2)$ It passes through the intersection of the lines $2x+y-8=0$ and $x-2y+1=0$, and has equal intercepts on the two coordinate axes.

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