
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
Given a rectangular prism $ABCD-A_{1}B_{1}C_{1}D_{1}$, where $AD=1$, $AB=2$, $A A_{1}=3$, and point $P$ is a point within plane $A_{1}BD$, find the minimum length of $AP$.

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