
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 an equilateral triangle \( \triangle ABC \) with \( AB = BC = CA = 5 \), vertex \( A \) is folded down to point \( A' \) on \( \overline{BC} \) such that \( BA' = 2 \) and \( A'C = 3 \). Find the length of crease \( \overline{PQ} \).

[asy]
unitsize(1 cm);

pair A, Ap, B, C, P, Q;

A = 5*dir(60);
B = (0,0);
C = (5,0);
Ap = (2,0);
P = (5*2/7)*dir(60);
Q = C + (5*3/8)*dir(120);

draw(B--C--Q--P--cycle);
draw(P--Ap--Q);
draw(P--A--Q,dashed);

label("$A$", A, N);
label("$A'$", Ap, S);
label("$B$", B, SW);
label("$C$", C, SE);
label("$P$", P, NW);
label("$Q$", Q, NE);
[/asy]

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