How much to rebalance into a double-or-half asset

Each year an asset either doubles or halves in value, with probability $1/2$ each, independently from year to year. Cash earns nothing. At the start of every year you rebalance so that a fraction $f$ of your wealth is in the asset and the rest is in cash.

(a) With $f = 1$, what is the expected one-year return? What is the long-run growth rate of a typical path?

(b) Which $f$ maximizes the long-run growth rate, and what is that rate?

(c) Over 100 years the asset doubles 50 times and halves 50 times. Compare your final wealth under $f = 1$ and under your answer to (b).