
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 $1$, let $P$ be a moving point on the space diagonal $B C_{1}$ and $Q$ be a moving point on the base $A B C D$. Find the minimum value of $D_{1} P + P Q$.

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