
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 thin homogeneous rod of length \( l \) and mass \( M \) is hinged at its upper end. A mass \( M_1 \) is attached to the other end of the rod via a hinge, where the hinge is exactly at the center of mass of the mass \( M_1 \). A bullet of mass \( m \) is fired horizontally with a velocity \( v \) and penetrates \( M_1 \) at the height of the hinge (without passing through to the axis). What will be the maximum displacement of the pendulum?

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