
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
For the equation in terms of $x$, $x^{2}-(6+i)x+9+ai=0$ ($a\in\mathbb{R}$) has a real root $x=b$.

$(1)$ Find the values of the real numbers $a$ and $b$.

$(2)$ If the complex number $z_{1}= \frac {2}{1+i}$, and the complex number $z$ satisfies $|z-a-bi|=|z_{1}|$, find the minimum value of the modulus $|z|$ of the complex number $z$.

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