HMMT 2014 U17

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$. (b) $f(a) \leq f(b)$ whenever $a$ and $b$ are positive integers with $a \leq b$. (c) $f(2 a)=f(a)+1$ for all positive integers $a$.

How many possible values can the 2014-tuple $(f(1), f(2), \ldots, f(2014))$ take?