Cute exponentials

Geometry Level 4

Solve \[\large \begin{cases} { e }^{ x+y }+{ e }^{ y+z }+{ e }^{ z+x }=1 \\ { e }^{ 2x }+{ e }^{ 2y }+{ e }^{ 2z }=\frac { 26 }{ 27 } +{ e }^{ 2x+2y+2z } \end{cases} \] If \( x+y+z=\)\( -\dfrac ba\ln { b } \), where \( a \) and \( b \) are primes, find \( a+b+1 \).

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