
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 $a\in R$, the complex numbers $z_{1}=a-1+i$, $z_{2}=1-\frac{{a+2}}{a}i$, and $z_{3}=-1$ correspond to points $A$, $B$, and $C$ on the complex plane, respectively, with $O$ as the origin.<br/>$(1)$ Find the range of $|z_{1}+z_{2}|$;<br/>$(2)$ When points $A$, $B$, and $C$ are collinear, find the area of triangle $AOB$.

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