
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
On a horizontal table surface α, there are four balls each with a radius of $2R$ placed such that they all touch each other (the connections between the centers of the balls form a square). A smaller ball with a radius of $R$ is placed on top of these four balls, and it just touches each of the four balls below. Find the distance from the center of the smaller ball to the horizontal table surface α.

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