I bet you can't solve this

Algebra Level 5

The function \(f : \mathbb{N} \to \mathbb{N}\) satisfies the condition \(f(m+n) \ge f(m)+f(f(n))-1\) for all \(m,n \in \mathbb{N}\). Find the sum of all possible values of \(f(2007)\).

Let this sum be \(k\). Submit your answer as the sum of digits of \(k\).

Assumption: \(\mathbb{N}\) is the set of all positive integers.

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