Implement inverted dropout for a NumPy array. During training, independently drop each element with probability p. Scale every retained element by $1 / (1 - p)$ so the expected output magnitude is unchanged.

Use rng.random(x.shape) when an rng generator is provided. Otherwise, use np.random.random(x.shape). The test panel displays a seed; the runner creates np.random.default_rng(seed) and passes that generator as rng.

Return (output, dropout_pattern), where dropout_pattern is the scaled mask applied to the input. Its entries are 0 for dropped elements and $1 / (1 - p)$ for retained elements.