An arithmetic progression of \(n\) terms has the following characteristics:

- The first term \(a\) is a positive odd integer
- The common difference \(d\) is a positive integer
- The terms of the progression alternate between odd and even integers.

Now, it is given that

\[\left(T_2 + T_4 + T_6 + T_8 + \ldots\right) - \left(T_1 + T_3 + T_5 + T_7 + \ldots\right) = 501\]

Find the value of \(p\) if \(n \equiv p \pmod{4}\)

Clarification: \(T_k\) represents the \(k\)-th term in the progression

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