
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

One hundred bear cubs found berries in the forest: the youngest managed to grab 1 berry, the next oldest grabbed 2 berries, the next took 4 berries, and so on; the oldest cub received $2^{99}$ berries. A fox proposed to divide the berries "fairly". She can approach two bear cubs and distribute their berries equally between them, and if there is an extra berry, the fox eats it. She continues these actions until all the bear cubs have an equal number of berries. What is the minimum number of berries the fox can leave for the bear cubs?

Present the answer in LaTex format: \boxed{Your answer}
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