Question 28
The sequence of triples of positive integers $(a_0, b_0, c_0)$, $(a_1, b_1, c_1)$, ..., $(a_{2025}, b_{2025}, c_{2025})$ satisfies
$$ \left(a_{i+1},b_{i+1},c_{i+1}\right)=\left(a_{i}+\gcd(b_{i},c_{i}),b_{i}+\gcd(c_{i},a_{i}),c_{i}+\gcd(a_{i},b_{i})\right) $$
for $ 0 \leq i \leq 2024 $.
Determine the smallest possible value of $a_{2025}$.
Here $ \gcd(x,y) $ denotes the greatest common divisor of integers x and y.
$$ \left(a_{i+1},b_{i+1},c_{i+1}\right)=\left(a_{i}+\gcd(b_{i},c_{i}),b_{i}+\gcd(c_{i},a_{i}),c_{i}+\gcd(a_{i},b_{i})\right) $$
for $ 0 \leq i \leq 2024 $.
Determine the smallest possible value of $a_{2025}$.
Here $ \gcd(x,y) $ denotes the greatest common divisor of integers x and y.