
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 fixed positive integer and fix a point  $O$  in the plane. There are  $n$  lines drawn passing through the point  $O$ . Determine the largest  $k$  (depending on  $n$ ) such that we can always color  $k$  of the  $n$  lines red in such a way that no two red lines are perpendicular to each other.

*Proposed by Nikola Velov*

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