I want to show that reasonable advice can really speed up computation.

Show, that every time-constructible function $t$, there exists a set $S$ in time $\text{DTIME}(t^2) \setminus \text{DTIME}(t)$ that can be decided in linear time using an advice of linear length, $S \in \text{DTIME}(l) / l$ (definition in analogy to $P / \text{poly}$), where $l(n)=O(n)$.

The problem is that I cannot come up with the structure of set.

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Hint: Start with a problem in $\mathrm{DTIME}(e^{2t}) \setminus \mathrm{DTIME}(e^t)$.