
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
A dragon has imprisoned a knight and given him 100 different coins, half of which are magical (only the dragon knows which ones). Each day, the knight divides all the coins into two piles (not necessarily equal). If both piles have the same number of magical coins or the same number of ordinary coins, the dragon will release the knight. Can the knight be guaranteed to be set free no later than

a) on the 50th day

b) on the 25th day

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