
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
The graph of the rational function $\frac{p(x)}{q(x)}$ is revealed below. Assume $q(x)$ is a quadratic expression, $p(1)=4$, and $q(3) = 0$. Also $q(x)$ has a double root at $x=3$. Find the expression for $p(x) + q(x)$.

[asy]
size(8cm);
import graph;

Label f; 
f.p=fontsize(6); 
draw((-5,0)--(5,0));
draw((0,-5)--(0,5));

int i;

for (i = -5; i <= 5; ++i) {
  if (i != 0) {
    draw((i,-0.2)--(i,0.2));
    draw((-0.2,i)--(0.2,i));
   label("$" + string(i) + "$", (i,-0.2), S);
   label("$" + string(i) + "$", (-0.2,i), W);
  }
}

real f(real x) {return (x-1)/(x-3)^2;}

draw(graph(f,-5,2.9));
draw(graph(f,3.1,5));
fill(circle((3,-1),0.15),white); // hole at x=3
[/asy]

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