\[f(x) = \begin{cases} x^3 + 2x^2 + x + c, & \text{if } x \leq b \\ e^x, & \text{if } x > b. \end{cases}\] Let a function \(f : \mathbb{R} \rightarrow \mathbb{R}\) be defined as the piecewise function above.

It is given that \(f\) is differentiable for all real values of \(x\), and \(b\) and \(c\) are integers.

Find \(b + c\).

**Clarification**: \(e\) denotes Euler's number, \(e \approx 2.71828\).

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