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E. Something's Fishy
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

Dylan Smith noticed something fishy with the blog post. In order to confirm his suspicions, he wants you to compute the fishy frequency constant, which corresponds to the floor of the hypotenuse $$$h$$$ of the right triangle with leg lengths:

$$$a = \lim_{k \rightarrow \infty} (\frac{\int_{0}^{\infty}2-\frac{2x^2}{1+x^2} dx}{1 - \frac{1}{k! + 1}})^{k!} \cdot e^{-k! \ln (\frac{1}{4}\int_{0}^{2\pi} \sqrt{2 + \cos x + \sqrt{5 + 4\cos x}} dx)}$$$
$$$b = \sum_{k=0}^{\infty} ( \frac{1}{(\sum_{j=0}^{\infty} \frac{16(-1)^j(2j)!}{(j!)^2(4j+1)4^{100j}})^k} (\frac{8k+7}{32k^2 + 20k + 2} - \frac{1 + \int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} dx}{8k+5} - \frac{4 \cdot \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}}{(8k+6) \cdot \int_{0}^{\infty} \frac{dx}{\sqrt{x} (x+1)}}))$$$
$$$h = \lfloor \sqrt{a^2 + b^2} \rfloor$$$
Output

Print one integer, the fishy frequency constant.