
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 different lines $l_{1}$: $ax+3y+1=0$, $l_{2}$: $x+(a-2)y+a=0$  
$(1)$ If $l_{1}\bot l_{2}$, find the value of the real number $a$;  
$(2)$ If $l_{1}$ ∥ $l_{2}$, find the value of the real number $a$ and determine the distance between the lines $l_{1}$ and $l_{2}$.

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