
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 two lines $l_{1}$: $ax+2y+6=0$ and $l_{2}$: $x+(a-1)y+a^{2}-1=0$  
$(1)$ Find the value of $a$ when $l_{1}\perp l_{2}$;  
$(2)$ Under the condition in $(1)$, if a line $l_{3}$ is parallel to $l_{2}$ and passes through point $A(1,-3)$, determine the general equation of line $l_{3}$.

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