Four regular hexagons with side length \(22\) are attached by their sides to a square, also with side length \(22\). The hexagons are then folded upwards to create a bowl. From base to rim, the height of this bowl can be expressed as \(\sqrt{a}\), where \(a\) is a positive integer. Find the value of \(a\).

**Details and assumptions**

From base to rim is the height of the bowl normal to the base, i.e straight up and down.

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