An outcome of flipping a coin $n$ times is called $k$-lucky if it contains a pattern which is repeated $k$ times in a row. For example, the outcome THHTHTHTTH (T stands for “tales” and H for “heads”) of flipping a coin 10 times is 3-lucky since HT appears 3 times in a row. Let $P_n$ be the probability that flipping a coin $n$ times is $6$-lucky. Find $t$ as small as you can so that $P_n<t$ for all $n$.