Let's solve the problem step by step.

Step 1: Understanding the Random Variable $X_i$

Given that an unbiased die is thrown 13 times, an ordered sequence of '+' (for outcomes 2 or 5) and '-' (for outcomes 1, 3, 4, or 6) is formed. For each $X_i$ (where $i = 1, 2, \ldots, 11$), the value of $X_i$ is determined as follows:

• $X_i = 1$ if both of its neighboring signs differ from the sign in between.
• $X_i = 0$ otherwise.

Step 2: Probability of Each Sign

Since the die is unbiased:

• Probability of a '+' (for outcomes 2 or 5) is $P(+) = \frac{2}{6} = \frac{1}{3}$.
• Probability of a '-' (for outcomes 1, 3, 4, or 6) is $P(-) = \frac{4}{6} = \frac{2}{3}$.

Step 3: Calculation of $P(X_i = 1)$

$X_i = 1$ occurs if the sign sequence follows either '+ - +' or '- + -'. The probability of each scenario is:

• $P(+ - +) = \frac{1}{3} \times \frac{2}{3} \times \frac{1}{3} = \frac{2}{27}$.
• $P(- + -) = \frac{2}{3} \times \frac{1}{3} \times \frac{2}{3} = \frac{4}{27}$.

Thus, the total probability that $X_i = 1$ is: $P(X_i = 1) = P(+ - +) + P(- + -) = \frac{2}{27} + \frac{4}{27} = \frac{6}{27} = \frac{2}{9}.$

Step 4: Expected Value of $S$

$S = \sum_{i=1}^{11} X_i$, where each $X_i$ is an independent random variable. The expected value of $S$ is: [ \mathbb{E}[S] = \sum_{i=1}^{11} \mathbb{E}[X_i] = 11 \times \mathbb{E}[X_i] = 11 \times \frac{2}{9} = \frac{22}{9}. ]

Step 5: Expected Value of $Y$

$Y = 14S + 29$. The expected value of $Y$ is: [ \mathbb{E}[Y] = 14 \times \mathbb{E}[S] + 29 = 14 \times \frac{22}{9} + 29 = \frac{308}{9} + 29 = \frac{308}{9} + \frac{261}{9} = \frac{569}{9}. ]

Thus, the expected value of $Y$ is $\boxed{\frac{569}{9}}$.