
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
In triangle $ABC$, which is a right triangle with $AB = AC$, points $P$, $Q$, and $R$ are the midpoints of sides $AB$, $BC$, and $CA$, respectively. Points $S$ is the midpoint of $PR$ and $T$ is the midpoint of $RQ$. Determine the ratio of the area of the shaded region (inside triangle $PST$) to the non-shaded region in triangle $ABC$.

[asy]
draw((0,0)--(1,0)--(0,1)--(0,0)--cycle,linewidth(1));
filldraw((0,0)--(1/2,1/2)--(1/2,0)--(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("A", (0,1), W);
label("B", (0,0), SW);
label("C", (1,0), E);
label("P", (0,1/2), W);
label("Q", (1/2,0), S);
label("R", (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}
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