
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 \(ABCD-A_1B_1C_1D_1\), \(AB = 2\sqrt{2}\), \(BC = AA_1 = 2\). Point \(P\) is a moving point on diagonal \(AC_1\), and point \(Q\) is a moving point on the base \(ABCD\) (points \(P\) and \(Q\) can coincide). Find the minimum value of \(B_1P + PQ\).

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