= Solution
For $\kappa\leq4$, part i gives no nonzero boundary intersection. For $\kappa>4$, part i gives a boundary hit almost surely. After any hit, map out the past hull and recenter at the current tip. The <Conformal Markov property of SLE> says that the future is again an <SLE> in the remaining domain. Applying part i repeatedly and using <Scaling invariance of SLE> produces another boundary hit after every finite number of hits. Therefore there are almost surely infinitely many.
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