
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 a line $l$ passes through the intersection point of two lines $l_{1}: 2x-y+4=0$ and $l_{2}: x-y+5=0$, and is perpendicular to the line $x-2y-6=0$.
$(1)$ Find the equation of line $l$;
$(2)$ If the distance from point $P(a,1)$ to line $l$ is $\sqrt{5}$, find the value of the real number $a$.

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