
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 $PQR$, the orthocenter divides altitude $\overline{PS}$ into segments with lengths $HS = 10$ and $HP = 25.$ Calculate $\tan Q \tan R.$

[asy]
unitsize (1 cm);

pair P, Q, R, G, S, H;

P = (0,0);
Q = (6,0);
R = (5,5);
G = (P + reflect(Q,R)*(P))/2;
S = (P + reflect(Q,R)*(P))/2;
H = extension(P,G,Q,S);

draw(P--Q--R--cycle);
draw(P--S);

label("$P$", P, SW);
label("$Q$", Q, SE);
label("$R$", R, N);
label("$S$", S, S);
dot("$H$", H, W);
[/asy]

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