Find the number of five-digit positive integers, $n$ , that satisfy the following conditions: 
$(a)$ the number $n$ is divisible by $5,$ 
$(b)$ the first and last digits of $n$ are equal, and 
$(c)$ the sum of the digits of $n$ is divisible by $5.$

Let's think step by step and output the final answer within \boxed{}.
user