\[ \large \lim_{n\to\infty} \sum_{r=1}^n \frac{\ln(n^2+r^2) - 2\ln(n)}{n} = \ln(2) + \frac\pi2-2 \]

We are given the value of the limit above. Suppose we consider the limit below

\[ \large \lim_{n\to\infty} \frac1{n^{2m}} \left[ (n^2+1)(n^2+2^2)(n^2+3^2)\ldots(n^2+n^2)\right]^{\frac mn} \]

If this limit equals to \( (ae^{b-a})^m \) for constant \(m\) and positive integer \(a\), find the value of \(b\).

**Clarification:** \(\displaystyle e= \lim_{L \to 0} (1+L)^{1/L } \).

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