Find all real $a,b,c,d,e,f$ that satisfy the system
$4a = (b + c + d + e)^4$
$4b = (c + d + e + f)^4$
$4c = (d + e + f + a)^4$
$4d = (e + f + a + b)^4$
$4e = (f + a + b + c)^4$
$4f = (a + b + c + d)^4$