
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
The function \( f \) assigns a number \( f(\boldsymbol{v}) \) to each vector \( \boldsymbol{v} \) (with a common origin at point \( O \) ) in space, such that for any vectors \( \boldsymbol{u}, \boldsymbol{v} \) and any numbers \( \alpha, \beta \), the value \( f(\alpha \boldsymbol{u} + \beta \boldsymbol{v}) \) does not exceed at least one of the numbers \( f(\boldsymbol{u}) \) or \( f(\boldsymbol{v}) \). What is the maximum number of values that such a function can take?

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