
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 $ABC,$ points $D$ and $E$ are on line segments $\overline{AB}$ and $\overline{AC}$, respectively, and angle bisector $\overline{AT}$ intersects $ \overline{DE}$ at $F.$ If $AD = 1,$ $DB = 4,$ $AE = 3,$ and $EC = 3,$ compute $\frac{AF}{AT}.$

[asy]
unitsize(1 cm);

pair A, B, C, D, E, F, T;

B = (0,0);
C = (7,0);
A = intersectionpoint(arc(B,4,0,180),arc(C,6,0,180));
D = interp(A,B,1/5);
E = interp(A,C,3/6);
T = extension(A, incenter(A, B, C), B, C);
F = extension(A, T, D, E);

draw(A--B--C--cycle);
draw(A--T);
draw(D--E);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, W);
label("$E$", E, NE);
label("$F$", F, SW);
label("$T$", T, S);
[/asy]

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