Consider a function \(f : \mathbb{R} \rightarrow \mathbb{R}\) defined as:

\[f(x) = \begin{cases}\displaystyle\int\limits_ {|x|} ^{x^2} \ln t~ dt & \text{if } x \leq a \\ \displaystyle\int\limits_ {a} ^{x} e^t \sin t~ dt & \text{if } x > a\end{cases}\]

Let \(a_1, a_2, a_3 ...... a_n\) be the values of \(a\) for which \(f\) is continuous throughout its domain.

Find

\[\displaystyle\prod_{i = 1} ^ n (a_i- 2)^2\]

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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