
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
Given is natural number  $n$ . Sasha claims that for any  $n$  rays in space, no two of which have a common point, he will be able to mark on these rays  $k$  points lying on one sphere. What is the largest  $k$  for which his statement is true? 

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