How do I construct a NTM that accepts the language consisting of the coding of turing machines that halt on one input?

Question

I currently have a problem with the following question:

Let $L = \{ \langle M \rangle \mid \exists w: \text{$M$ halts for $w$ in at most $|w|^3$ steps} \}$. Construct an NTM (non-deterministic Turing machine) that decides $L$.

My idea was to simulate every possible input of length $\le|w|^3$ on a given TM $M$ by using a separate band in my NTM for every single input word. If there is one band where $M$ halts, then the NTM accepts.

Is this the right way to go about it? My problem is that I don't see why I would need a NTM to decide $L$. A standard TM would be able to do the same thing.