
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 one hundred cards numbered from 1 to 100. They place each card into one of three boxes - a red box, a white box, and a blue box - in such a way that each box contains at least one card.

An audience member selects two of the three boxes and takes one card from each. They then announce the sum of the numbers on the two cards. Knowing this sum, the magician can determine from which box no card was taken.

How many ways are there to distribute the cards into the boxes so that this trick always works? (Two distributions are considered different if at least one card is placed into a different box.)

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