Inverse trigonometry or number theory

Geometry Level 4

\[\large \text{arccot} (8)=\arctan (a)-\arctan(b)\]

If the ordered pair of natural numbers \((a_1,b_1)\) and \((a_2,b_2)\) satisfy the equation above, find the minimum value of \(a_1+b_1+a_2+b_2\).