
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
The edge length of the cube $ABCD-A_1B_1C_1D_1$ is $1$. If the moving point $P$ is on the line segment $BD_1$, then the range of values for $\overrightarrow{DC} \cdot \overrightarrow{AP}$ is .

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