
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
Solve for the real numbers $a$ and $b$ such that the equation $x^2 - (6+i)x + 9 + ai = 0$ ($a \in \mathbb{R}$) has a real root $x=b$.  
(1) Determine the values of real numbers $a$ and $b$.  
(2) Let the complex number $z_1 = \frac{2}{1+i}$, and another complex number $z$ satisfy the condition $|z - a - bi| = |z_1|$. Find the minimum value of the modulus $|z|$.

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