
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 side of an equilateral triangle is divided into \( n \) equal segments, and lines parallel to the sides are drawn through all the division points. The triangle is thus divided into \( n^2 \) smaller triangular cells. The triangles located between two neighboring parallel lines form a strip.

a) What is the maximum number of cells that can be marked such that no two marked cells belong to the same strip in any of the three directions, if \( n=10 \)?

b) The same question for \( n=9 \).

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