
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
What is the maximum number of cells that can be crossed by:

a) A line segment drawn on a piece of "grid" paper of size \( m \times n \) cells;

b) A needle of length 200 (with a cell side being the unit of length) thrown onto a piece of "grid" paper? 

[The segment (or "needle") is considered to be crossing a cell if it passes through its interior point.]

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