
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
A cube \( ABCD-A_1B_1C_1D_1 \) with edge length 2 moves in a 3D Cartesian coordinate system \( O-xyz \). The vertex \( A \) remains on the \( z \)-axis, and the vertex \( B_1 \) remains on the \( xOy \) plane. Find the minimum length of \( OC \).

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