
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
Find $AX$ in the diagram given below:

[asy]
import markers;
real t=.65;  // Change the division point for creating a new ratio

pair A=(0,0);
pair B=(4,3);  // Changed coordinates for B to make new lengths
pair C=(1,2);  // Changed coordinates for C
pair X=t*A+(1-t)*B;  // X is the division point recalculated with new t

draw(C--A--B--C--X);
label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$X$",X,SE);

markangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1),true)));
markangle(n=1,radius=15,X,C,B,marker(markinterval(stickframe(n=1),true)));

label("$30$", .5*(B+X), SE);  // New distance BX
label("$36$", .5*(B+C), N);  // New distance BC
label("$27$", .5*(A+C), NW);  // New distance AC
[/asy]

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