Telling fibs

Calculus Level 5

Let the matrix with entries $$a_{ij}$$ be defined as

$\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \exp \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$

$a_{12}+a_{21} = \frac{2}{\sqrt{d}}(e^{p}-e^{-1/p})$

and $$p = \frac{a+\sqrt{b}}{c}$$, where $$a,b,c,$$ and $$d$$ are positive integers, find $$a+b+c+d$$.

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