How often does a trend follower lose?

An asset's price, measured in price units from where it starts, is $X_t = \mu t + W_t$ for $t \in [0, 1]$, where $W$ is a standard Brownian motion. A trend follower holds $X_t$ units at time $t$: long when the price is above its starting level, short when it is below, rebalancing continuously. Each position is set before the next price move, so the P&L is the Itô integral

(a) Write $\Pi$ in terms of $X_1$ alone, and find $E[\Pi]$ as a function of $\mu$.

(b) Take $\mu = 0$. What is the probability that the strategy loses money? What is the largest possible loss? What are the average loss given a loss and the average gain given a gain?

(c) Now the asset trends, with $\mu = 1$ or $\mu = -1$. What are $E[\Pi]$ and the probability of a loss?