
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
A billiard table is in the shape of a $2 \times 1$ rectangle, with pockets located at the corners and midpoints of the long sides. What is the minimum number of balls that need to be placed inside the rectangle so that each pocket is collinear with some two balls?

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