Let $k \geq 1$ be an integer. We consider $4 k$ chips, $2 k$ of which are red and $2 k$ of which are blue. A sequence of those $4 k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r \underline{b b b r} \underline{r} \underline{b} b$ to $r \underline{r} \underline{b} b \underline{b b b} b$ where $r$ denotes a red chip and $b$ denotes a blue chip.

Determine the smallest number $n$ (as a function of $k$ ) such that starting from any initial sequence of the $4 k$ chips, we need at most $n$ moves to reach the state in which the first $2 k$ chips are red.