
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 one managed to grab 1 berry, the next oldest cub 2 berries, the next 4 berries, and so on, the oldest one got $2^{99}$ berries. A fox proposed to "fairly" distribute the berries among all the cubs. She can approach any two cubs and evenly distribute their berries between them, and if an extra berry is left over, the fox eats it. She continues these actions until all 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}
user