
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
Let \( n \) be a positive integer, and denote the plane point set as
$$
S = \{(x, y) \mid x + y \in \{0, 1, \cdots, n\}, x + y \neq 0 \}.
$$

Question: What is the minimum number of lines required, such that the union of these lines covers \( S \) but does not include the point \( (0, 0) \)?

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