Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: Let $A$ and $B$ be two points on the parabola $y = x^2,$ such that when the tangents at $A$ and $B$ drawn, they are perpendicular.  Then for any such pair of tangents, the $y$-coordinate of their point of intersection $P$ is always the same.  Find this $y$-coordinate.

[asy]
unitsize(1.5 cm);

real parab (real x) {
  return(x^2);
}

pair A, B, P;
real a, b;

a = 1;
b = (-1/4)/a;
A = (a,a^2);
B = (b,b^2);
P = extension(A, A + (1,2*a), B, B + (1,2*b));

draw(graph(parab,-1.5,1.5));
draw(interp(A,P,-0.8)--interp(A,P,1.2));
draw(interp(B,P,-1)--interp(B,P,1.5));
draw(rightanglemark(A,P,B,5));

label("$P$", P, S);

dot("$A$", A, SE);
dot("$B$", B, S);
[/asy]
user