Consider the following experiment. Step 1. Flip a fair coin twice. Step 2. If the outcomes are (TAILS, HEADS) then output $Y$ and stop. Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output $N$ and stop. Step 4. If the outcomes are (TAILS, TAILS), then go to Step 1. The probability that the output of the experiment is $Y$ is (up to two decimal places) ________ .

When a fair coin is flipped twice, there are 4 equally likely outcomes, each with probability $\frac{1}{4}$:

1. $(\text{TAILS}, \text{HEADS})$ with probability $\frac{1}{4} \implies$ Output $Y$ and stop.

2. $(\text{HEADS}, \text{HEADS})$ or $(\text{HEADS}, \text{TAILS})$ with total probability $\frac{2}{4} = \frac{1}{2} \implies$ Output $N$ and stop.

3. $(\text{TAILS}, \text{TAILS})$ with probability $\frac{1}{4} \implies$ Repeat Step 1.

Let $P(Y)$ be the probability that the experiment eventually outputs $Y$. By the law of total probability:

Solving for $P(Y)$: