Sum of sum

The function \(f(x)\) is the sum of digits of \(x\); for example: \(f(24)=6\), \(f(78)=15\), and \(f(144)=9\) (note that \(f(99)=18\) and not \(1+8=9\)).

Find \(f(1)+f(2)+f(3)+ \cdots + f(1000)\).

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