
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 15. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance 4 from point $C$. Find the square of the length of the line segment along which the triangle is folded.

[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", (11,0), dir(-80));
pair C = MP("C", (15,0), dir(-20));
pair K = (7.5,12.990);
pair M = (a*B+(15-a)*K) / 15;
pair N = (b*C+(15-b)*K) / 15;
draw(B--M--N--C--cycle, tpen);
fill(M--A--N--cycle, mediumgrey);
draw(M--A--N--cycle);
pair shift = (-20.73, 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}
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