Solving Circular Absolute Difference Equations with a Fixed Sum

Six numbers, denoted as $a_1, a_2, a_3, a_4, a_5, a_6$, are arranged in a circle. Each number is defined by the equation:

for $i = 1, 2, 3, 4, 5, 6$, where the indices are taken modulo 6 (i.e., $a_7 = a_1$ and $a_8 = a_2$). Additionally, the sum of all the numbers is equal to 1:

Determine the values of these six numbers.