
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 lines $l_{1}$: $\left(2a+1\right)x+\left(a+2\right)y+3=0$ and $l_{2}$: $\left(a-1\right)x-2y+2=0$.  
$(1)$ If $l_{1}$ is parallel to $l_{2}$, find the value of $a$.  
$(2)$ If $l_{1}$ is perpendicular to $l_{2}$, find the value of $a.

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