
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 right trapezoid $ABCD$, $AB \parallel DC$, $AD \perp AB$, $AD = DC = 2$, $AB = 3$. Let point $M$ be a movable point inside the trapezoid $ABCD$ (including the boundary), and let point $N$ be the midpoint of side $CD$. Find the maximum value of $\overrightarrow{AM} \cdot \overrightarrow{AN}$.

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