Alice has two boxes \(A\) and \(B\). Initially box \(A\) contains \(n\) coins and box \(B\) is empty. On each turn, she may either move a coin from box \(A\) to box \(B\), or remove \(k\) coins from box \(A\), where \(k\) is the current number of coins in box \(B\) . She wins when box \(A\) is empty.

\((a)\) If initially box \(A\) contains \(6\) coins, show that Alice can win in \(4\) turns.

\((b)\) If initially box \(B\) contains \(31\) coins, show that Alice cannot win in \(10\) turns.

\((c)\) What is the minimum number of turns needed for Alice to win if box \(A\) initially contains \(2018\) coins?