Think of it the other way!

Algebra Level 4

Consider the following system of linear equations:

\[\begin{cases} \begin{array}{rcrcrcrcrcrcrl} 1&+&a&+&b&+&c&+&d&+&e&=&1\\ 32&+&16a&+&8b&+&4c&+&2d&+&e&=&2\\ 243&+&81a&+&27b&+&9c&+&3d&+&e&=&3\\ 1024&+&256a&+&64b&+&16c&+&4d&+&e&=&4\\ 3125&+&625a&+&125b&+&25c&+&5d&+&e&=&5&.\\ \end{array} \end{cases}\]