Four positive integers \(a\), \(b\), \(c\) and \(d\) are such that \[abcd+abc+bcd+cda+dab+ab+bc+cd+da+ac+bd+a+b+c+d=2009.\]

What is the value of \(a+b+c+d\)?

Can we (almost) factorise the left hand side?

Or can we add something to the left hand side that means we can factorise it perfectly?