It's a toss-up ....

A game is played with a fair coin where you are asked to toss the coin \(10\) times. If you make it through the \(10\) tosses without tossing a sequence of either \(3\) (or more) consecutive heads or \(3\) (or more) consecutive tails in a row then you win \($5\), otherwise you pay \($1\).

Your expected monetary return, (in dollars), after playing this game \(100\) times is \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

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