This sounds fun, until you realize your life is at stake

Discrete Mathematics Level 5

In the year 4096, while making an intergalactic excursion to planet 2048 in galaxy 1024, your space vehicle ran out of fuel and you crash landed on planet 512. Unfortunately, the inhabitants of​ the planet do not welcome visitors and so planned to execute you by chopping you into 256 pieces before scattering your remains to 128 different national parks to be fed on by 64 different strange creatures residing in each park.

You have one chance to stay alive: Before the day of execution, you will participate in the "Powers of Two Stacking Festival". If you manage to emerge victorious in the festival, your life will be spared as the inhabitants think it would be a waste to execute such an intelligent living being.

At the start of the festival, all participants are placed in separate rooms where there are a countless number of cards with the number 2 written on them. Whenever two cards of 2 are stacked together, they magically fuse together to become a card of 4 and you are 'awarded' 4 points. In turn, when you stack together two cards of 4, they fuse to become a card of 8 and you are 'awarded' 8 points. As a general rule of thumb, two identical cards of \(2^{n}\) fuse to become a single card of \(2^{n+1}\) when they are stacked together and you are 'awarded' \(2^{n+1}\) points whenever you perform this operation.

The game ends when a card of 2048 is produced. Your final score is calculated by adding all the points 'awarded' to you in the process plus the number of operations you carry out (An operation is defined by the process of stacking together two cards, so you can only stack 2 cards in a single operation). All the participants are competing to get the lowest final score. The participant with the lowest final score is declared the winner. It is possible for multiple winners to exist if more than one participant share the same lowest final score.

To emerge victoriously, you need to achieve the least possible score allowed since some of the participants are extremely smart and will always achieve the least attainable score.

What is the minimum score attainable in the game?

This problem is inspired by the game 2048.

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