
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
In the cube \( ABCD - A_1B_1C_1D_1 \), points \( M \) and \( N \) are on segments \( AB \) and \( B B_1 \) respectively (excluding the endpoints of the segments), and \( AM = B_1N \). What is the range of the angle between \( A_1M \) and \( C_1N \)?

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