
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 the diagram, $P$ is on line segment $RS$ such that $QP$ bisects $\angle SQR$. $PQ=PR$, $\angle RSQ=3y^\circ$, and $\angle RPS=4y^\circ$. There is also a point $T$ on segment $RS$ such that $QT$ is perpendicular to $RS$. What is the measure, in degrees, of $\angle RPQ$? [asy]
import olympiad;
size(8cm);

real x = 45; real y = 20;

pair q = (1, 0);
pair r = (0, 0);
pair p = intersectionpoints((10 * dir(x))--r, q--(shift(q) * 10 * dir(180 - x)))[0];
pair s = intersectionpoints(r--(r + 10 * (p - r)), 10 * dir(180 - 3 * x)--q)[0];
pair t = (r + (s - r) * 0.5); 

// Draw lines
draw(p--s--q--p--r--q);
draw(q--t);

// Label points
label("$R$", r, SW);
label("$Q$", q, SE);
label("$S$", s, N);
label("$P$", p, NW);
label("$T$", t, S);

// Label angles
label("$y^\circ$", q, 4 * NW);
label("$y^\circ$", q, N);
label("$3y^\circ$", s, 4 * SW);
label("$4y^\circ$", p, 4 * S);
label("$90^\circ$", t, NW);

// Tick marks
add(pathticks(r--p, 2, spacing=0.6, s=2));
add(pathticks(p--q, 2, spacing=0.6, s=2));
add(rightanglemark(q, t, s));

[/asy]

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