A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon, $5$ centimeters of rain falls in the open field, partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.

Let's think step by step and output the final answer within \boxed{}.
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