# Piecewise Functions - 7

Algebra Level 4

$f(x) = \begin{cases} x^4 + 6x^3 -5x^2 -42x+40 & \text{if } x \leq a \\ \\ \frac{x}{2\pi} + \sin x& \text{if } x > a \end{cases}$

Let a function $$f : \mathbb{R} \rightarrow \mathbb{R}$$ be defined as above. As $$a$$ varies over all real numbers, the maximum number of distinct real roots of $$f(x) = 0$$ are $$N$$.

Let $$a_1, a_2, a_3, \ldots a_n$$ be the integer values of $$a$$ for which $$f(x) = 0$$ has $$N$$ distinct real roots.

Evaluate $$\displaystyle\sum_{i=1} ^{n} a_i$$.

Note: If you think there are no values of $$a$$ satisfying the conditions, enter your answer as 999.

This problem is part of the set - Piecewise-defined Functions

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