
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 triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .

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