
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
The lines  $AD , BE$  and  $CF$  intersect in  $S$  within a triangle  $ABC$  . 
It is given that  $AS: DS = 3: 2$  and  $BS: ES = 4: 3$  . Determine the ratio  $CS: FS$  .

[asy]
unitsize (1 cm);

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

A = (0,0);
B = (5,0);
C = (1,4);
S = (14*A + 15*B + 6*C)/35;
D = extension(A,S,B,C);
E = extension(B,S,C,A);
F = extension(C,S,A,B);

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

dot(" $A$ ", A, SW);
dot(" $B$ ", B, SE);
dot(" $C$ ", C, N);
dot(" $D$ ", D, NE);
dot(" $E$ ", E, W);
dot(" $F$ ", F, dir(270));
dot(" $S$ ", S, NE);
[/asy]

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