
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 displayed below. Assume $q(x)$ is a cubic polynomial, $p(4)=4$, and $q(3) = 3$. Determine $p(x) + q(x)$.

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

Label f; 
f.p=fontsize(6); 
//xaxis(-5,5,Ticks(f, 1.0)); 
//yaxis(-5,5,Ticks(f, 1.0));
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/((x-1)*x*(x+1))}

draw(graph(f,-5,-3.5), dashed);
draw(graph(f,-3.5,-1.1));
draw(graph(f,-0.9,-0.1));
draw(graph(f,0.1,0.7));
draw(graph(f,0.7,0.8), dashed);
draw(graph(f,1.2,1.3), dashed);
draw(graph(f,1.3,3.5));
draw(graph(f,3.5,5), dashed);
filldraw(circle((0,-1),.15),white);
[/asy]

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