
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 that $2+ai$ and $b+i$ (where $a, b \in \mathbb{R}$) are the roots of a quadratic equation with real coefficients $x^2+px+q=0$.
(1) Find the values of $a$, $b$, $p$, and $q$;
(2) Calculate: $\frac{a+bi}{p+qi}$.

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