
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
Each of the three people places their purple block in one of three empty boxes while the other two people can place their entire set of colored blocks into the remaining six boxes in $\binom{6}{3}^{2}$ equally likely ways, and the other blocks can be distributed into each of the remaining 5 boxes so that at least one box receives three blocks all of the same color is equal to $6 \cdot \binom{4}{1} \cdot \binom{5}{2} + 6 \cdot \binom{4}{2} \cdot \binom{3}{2} + 6 \cdot 4 + 3$.

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