Having trained in the finest restaurants, Chef Jean knows the optimal way to cook oven chips.

Ten chips are arranged on a tray, and placed in a hot oven. Each chip has two sides and must be cooked on both, so midway through the cooking, the tray is removed from the oven and each chip is turned over. If the chips stay out of the oven for too long, they go soggy, so for the crispiest chips, this turning must be done as quickly as possible.

There are two ways in which Jean can turn over chips, each of which takes exactly one second:

1. Shake the tray of chips, which flips over each chip independently with probability \(1/2\).
2. Turn over a single chip with tongs.

Each second, Jean either shakes the tray or turns over a single chip, choosing whichever action minimises the expected time until all the chips are the opposite way up from when they were taken out of oven. When all the chips are inverted in this way, the tray is returned to the oven.

Giving your answer in seconds, to three decimal places, what is the expected time that Jean takes to invert all the chips?

Mathsbombe Competition 2020 is organised by the The Department of Mathematics at The University of Manchester.
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