Ryski was asked compute the sum of the series

$\sum\limits_{k=0}^{∞}(1/((4k + 1)(4k + 2)(4k + 3)(4k + 4)))$.

Let this summation be $p$.

Ryski also has 2*$n$ different pens ($n$ ∈ $N$, $n$ ≥ 2). Each day, he takes $n$ pens with him to school. After some days the following condition was fulfilled: every two pens were together on at least one day. Let the minimum number of days needed for this to happen be $q$.

Give the answer as [$p$ * $q$ * ${10^{20}}$], where [$x$] denotes the greatest integer less than or equal to $x$.