
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 complex number $z=a+i\left(a\in R\right)$, where $i$ is the imaginary unit.<br/>$(Ⅰ)$ If $|z|=1$, find the value of $a$;<br/>$(Ⅱ)$ If $\frac{z}{{1+i}}$ is a real number, find the value of $a$;<br/>$(Ⅲ)$ If $z$ is a complex root of the real coefficient equation $x^{2}+bx+2=0$, find the values of $a$ and $b$.

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