
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
Two circles are centered at the origin. The point $P(5, 12)$ is on the larger circle and the point $S(0, k)$ is on the smaller circle. If $QR = 4$, what is the value of $k$?

[asy]
unitsize(0.2 cm);

defaultpen(linewidth(.7pt)+fontsize(10pt));
dotfactor=4;
draw(Circle((0,0),9)); draw(Circle((0,0),13));
dot((0,0)); dot((9,0)); dot((13,0)); dot((0,9)); dot((5,12));
draw((0,0)--(5,12));
label("$S (0,k)$",(0,9.5),W);
draw((15,0)--(0,0)--(0,15),Arrows(TeXHead));
draw((-15,0)--(0,0)--(0,-15));

label("$x$",(15,0),E); label("$y$",(0,15),N); label("$P(5,12)$",(5,12),NE);

label("$O$",(0,0),SW); label("$Q$",(9,0),SW); label("$R$",(13,0),SE);

[/asy]

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