
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
In the rectangular prism \(ABCD - A_{1}B_{1}C_{1}D_{1}\), the edge lengths are \(AB = 6\) and \(BC = BB_{1} = \sqrt{2}\). Point \(P\) is a moving point on the line segment \(BC_{1}\). What is the minimum value of \(AP + PB_{1}\)?

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