does any maximal ascending chain of prime ideals with the same ending prime ideal in a Noetherian integral domain have the same length?

Can we prove the following statement:

Let AA be a Noetherian integral domain and p\mathfrak{p} be a prime ideal. If the chain of prime ideals, 0=p0⊊p1⊊⋯⊊pn=p0=\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq\cdots\subsetneq\mathfrak{p}_n=\mathfrak{p}, is maximal (that is, we can not insert any prime ideals into this chain), then height \mathfrak{p}=n\mathfrak{p}=n.

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