
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 trapezoid PQRU depicted, the ratio of the area of triangle PQR to the area of triangle PUR is $5:2$. If $PQ + RU = 180$ cm, determine the length of segment $\overline{PQ}$. 
[asy]
import olympiad; size(150); defaultpen(linewidth(0.8));
pair P = (0,0), Q = (6,0), R = (4,3), U = (1.5,3);
draw(P--Q--R--U--cycle--R);
label("$P$", P, SW); label("$Q$", Q, SE); label("$R$", R, NE); label("$U$", U, NW);
[/asy]

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