
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
A paper equilateral triangle  $ABC$  has side length  $12$ . The paper triangle is folded so that vertex  $A$  touches a point on side  $\overline{BC}$  a distance  $9$  from point  $B$ . The length of the line segment along which the triangle is folded can be written as  $\frac{m\sqrt{p}}{n}$ , where  $m$ ,  $n$ , and  $p$  are positive integers,  $m$  and  $n$  are relatively prime, and  $p$  is not divisible by the square of any prime. Find  $m+n+p$ . 
[asy]
import cse5;
size(12cm);
pen tpen = defaultpen + 1.337;
real a = 39/5.0;
real b = 39/7.0;
pair B = MP("B", (0,0), dir(200));
pair A = MP("A", (9,0), dir(-80));
pair C = MP("C", (12,0), dir(-20));
pair K = (6,10.392);
pair M = (a*B+(12-a)*K) / 12;
pair N = (b*C+(12-b)*K) / 12;
draw(B--M--N--C--cycle, tpen);
draw(M--A--N--cycle);
fill(M--A--N--cycle, mediumgrey);
pair shift = (-20.13, 0);
pair B1 = MP("B", B+shift, dir(200));
pair A1 = MP("A", K+shift, dir(90));
pair C1 = MP("C", C+shift, dir(-20));
draw(A1--B1--C1--cycle, tpen);[/asy]

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