
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 \( q(x) \) is a quadratic polynomial and \( p(x) \) is a linear polynomial, with the graph of \( \frac{p(x)}{q(x)} \) having a vertical asymptote at \( x=1 \), a hole at \( x=0 \), and \( p(1)=1 \), \( q(3) = 3 \), find \( p(x) + q(x) \).

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