Let the (7,4) Hamming code be the 2-ary code obtained by encoding map
\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mapsto\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{1} \oplus x_{2} \oplus x_{4}, x_{1} \oplus x_{3} \oplus x_{4}, x_{2} \oplus x_{3} \oplus x_{4}\right)
where \oplus denotes addition modulo 2. Find, with proof, the minimal distance of this code.