
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 $l_{1}$: $x+a^{2}y+1=0$ and $l_{2}$: $(a^{2}+1)x-by+3=0$ where $a, b \in \mathbb{R}$.<br/>$(1)$ When $b=-2$, if $l_{1}$ <span class="special-symbol">∥</span> $l_{2}$, find the value of the real number $a$;<br/>$(2)$ If $l_{1}\bot l_{2}$, find the minimum value of $|ab|$.

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