
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 $PQR,$ points $M$ and $N$ are on $\overline{PQ}$ and $\overline{PR},$ respectively, and angle bisector $\overline{PS}$ intersects $\overline{MN}$ at $T.$ If $PM = 2,$ $MQ = 6,$ $PN = 4,$ and $NR = 8,$ compute $\frac{PT}{PS}.$

[asy]
unitsize(1 cm);

pair P, Q, R, M, N, T, S;

Q = (0,0);
R = (5.7,0);
P = intersectionpoint(arc(Q,3,0,180),arc(R,5,0,180));
M = interp(P,Q,2/8);
N = interp(P,R,4/12);
S = extension(P, incenter(P,Q,R), Q, R);
T = extension(P, S, M, N);

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

label("$P$", P, N);
label("$Q$", Q, SW);
label("$R$", R, SE);
label("$M$", M, SW);
label("$N$", N, NE);
label("$T$", T, NE);
label("$S$", S, S);
[/asy]

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