
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
Find the area of the pentagon bounded by the lines \( BC, CD, AN, AM \), and \( BD \), where \( A, B, \) and \( D \) are three vertices of the square \( ABCD \), \( N \) is the midpoint of side \( BC \), and \( M \) divides side \( CD \) in the ratio \( 2:1 \) (counting from vertex \( C \)), if the side length of the square \( ABCD \) is \( a \).

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