A geometry problem by Worranat Pakornrat

Geometry Level 4

Consider all sets of 4 points \(A, B, C, D \) which satisfy the following conditions:

\(AB\) is an integer.
\(BC = AB + 1 \).
\(CD = BC + 1\).
\( DA = CD + 1\).
\(AC= DA + 1\)
\( AC \) divides \( AB \times CD + BC \times DA \).

Over all such sets, what is \( \max \lceil BD \rceil ? \)