The four digit number \(n = a b c d\) satisfies the following properties:
(1) \(n \equiv 3 ( \bmod 7 )\)
(2) \(n\) is divisible by 9
(3) the first two digits have the same sum as the last two digits
(4) the digit \(b\) is smaller than any other digit
(5) the digit \(c\) is even
Use property (1) to explain why \(6 a + 2 b + 3 c + d \equiv 3 ( \bmod 7 )\)
Use properties (2), (3) and (4) to show that \(a + b = 9\)
Deduce that \(c \equiv 5 ( a - 1 ) ( \bmod 7 )\)
Hence determine the number \(n\), verifying that it is unique. You must make your reasoning clear.