
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
Point  $G$  is where the medians of the triangle  $ABC$  intersect and point  $D$  is the midpoint of side  $BC$ . The triangle  $BDG$  is equilateral with side length 1. Determine the lengths,  $AB$ ,  $BC$ , and  $CA$ , of the sides of triangle  $ABC$ .

[asy]
size(200);
defaultpen(fontsize(10));
real r=100.8933946;
pair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);
draw(A--B--C--A--D^^B--E^^C--F);
pair point=G;
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $D$ ", D, dir(point--D));
label(" $E$ ", E, dir(point--E));
label(" $F$ ", F, dir(point--F));
label(" $G$ ", G, dir(20));
label("1", B--G, dir(150));
label("1", D--G, dir(30));
label("1", B--D, dir(270));[/asy]

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