
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 the complex number $z=\frac{(1+i)^{2}+2(5-i)}{3+i}$.

(1) Find the modulus of $z$ ($\left| z \right|$) and determine the quadrant in which the point corresponding to the complex number $z$ lies in the complex plane.

(2) If $z(z+a)=b+i$, find the values of the real numbers $a$ and $b$.

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