
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 lines $l\_1$: $x+a^{2}y+1=0$ and $l\_2$: $(a^{2}+1)x-by+3=0$.
(1) If $b=-12$, $l\_1$ is parallel to $l\_2$, find the value of $a$.
(2) If $l\_1$ is perpendicular to $l\_2$, find the minimum value of $|a\cdot b|$.

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