Solution

In each word, a number is hidden and scrambled. For example, exist has the number six in it. All the numbers from these words are six, one, three, five. This gives the word "face" which triggers a hint saying: “Correct! No need to count on that anymore.” This alludes to the fact that the letters used to create numbers can now be discarded. If one takes the remaining letters from each word and smashes them together, one get tesgtary. Solving this anagram yields the password, strategy.