
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
We have 27 cubes with edges of $1 \mathrm{~cm}$.

a) Is it possible to paint these cubes' faces purple, white, and yellow in such a way that it is possible to assemble a $3 \times 3 \times 3$ cube with purple, white, and yellow surfaces?

b) What is the minimum number of faces that need to be painted purple to ensure that a $3 \mathrm{~cm}$ cube with a purple surface can be assembled?

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