
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 cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, point $P$ is on the edge $BC$, and point $Q$ is the midpoint of the edge $CC_{1}$. If the plane passing through points $A$, $P$, and $Q$ intersects the cube to form a pentagon, what is the range of values for $B P$?


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