= Solution
Consider an occurrence counted by $G_m$. Before the next counted occurrence, the exploration must first take a downward step of the <simple symmetric random walk>, which has probability $1/2$, and then choose the decrement $-1$ among the three equally likely values of $\xi$, which has probability $1/3$. Thus it terminates the visits to the current record value with probability
$$
p=\frac12\frac13=\frac16.
$$
The <Strong Markov property> at successive counted occurrences makes these trials independent. Therefore $G_m$ has the <geometric distribution> on $\{1,2,\ldots\}$ with parameter $1/6$, and
$$
\boxed{\mathbb P(G_m\geq j)=\left(1-\frac16\right)^{j-1}}
$$
for every $j\geq1$.
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