The line $y = mx$ bisects the angle between the two lines shown below. Find $m$.

[asy]
import markers;

unitsize(3 cm);

pair O, A, B, C;

O = (0,0);
A = dir(45 - 2*15);
B = dir(45);
C = dir(45 - 15);

draw((-0.2,0)--(1,0));
draw((0,-0.2)--(0,1));
draw((-0.2)*(A)--A);
draw((-0.2)*(B)--B);
draw((-0.2)*(C)--C);

label("$x$", (1,0), NE);
label("$y$", (0,1), NE);
label("$y = \frac{1}{7} x$", A, E);
label("$y = mx$", C, E);
label("$y = x$", B, NE);

markangle(1, A, (0,0), C, radius=10mm, marker(markinterval(stickframe(n=1,2mm),true)));
markangle(1, C, (0,0), B, radius=10mm, marker(markinterval(stickframe(n=1,2mm),true)));
[/asy]