
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 that the cuboid $ABCD-A_1B_1C_1D_1$ has dimensions where $DC + CC_1 = 8$, $CB = 4$, and $\overrightarrow{AM} = \overrightarrow{MB}$, and point $N$ is on the plane $A_1B_1C_1D_1$ with $C_1N = \sqrt{5}$, find the minimum value of the segment $MN$ when the volume of the cuboid $ABCD-A_1B_1C_1D_1$ is maximized.

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