
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 magician has 100 cards, each numbered from 1 to 100. He places these 100 cards into three boxes: one red, one white, and one blue. Each box contains at least one card. An audience member selects two of the three boxes and picks one card from each, then announces the sum of the numbers on these two cards. Knowing this sum, the magician is able to identify the box from which no card was chosen. How many ways are there to distribute the cards into the boxes such that the magician can always successfully identify the unchosen box? (Two distributions are considered different if at least one card is placed into a different color box.)

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