
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 that the lines $l_{1}$ and $l_{2}$ are perpendicular to each other and intersect at point $P(1,2)$.  
$(1)$ If the slope of $l_{1}$ is $2$, the intersection point of $l_{2}$ with the $x$-axis is $Q$, and point $M(a,b)$ moves on the line segment $PQ$, find the range of $\frac{{b-1}}{{a+1}}$.  
$(2)$ If the lines $l_{1}$ and $l_{2}$ intersect the $y$-axis at points $A$ and $B$ respectively, find the minimum value of $|AB|$.

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