
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
Triangle $PQR$ is a right isosceles triangle. Points $M$, $N$, and $O$ are the midpoints of sides $PQ$, $QR$, and $RP$, respectively. Point $S$ is the midpoint of segment $MO$ and point $T$ is the midpoint of segment $ON$. Determine the ratio of the shaded area to the non-shaded area in triangle $PQR$ given that the shaded area includes triangle $MNO$ and the non-shaded area includes the rest of triangle $PQR$.

[asy]
draw((0,0)--(1,0)--(0,1)--(0,0)--cycle,linewidth(1));
filldraw((0,0)--(1/2,0)--(1/2,1/2)--(0,1/2)--(0,0)--cycle,gray, linewidth(1));
filldraw((1/2,0)--(1/2,1/4)--(1/4,1/2)--(0,1/2)--(1/2,0)--cycle,white,linewidth(1));

label("P", (0,1), W);
label("Q", (0,0), SW);
label("R", (1,0), E);
label("M", (0,1/2), W);
label("N", (1/2,0), S);
label("O", (1/2,1/2), NE);
label("S", (1/4,1/2), N);
label("T", (1/2,1/4), E);
[/asy]

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