
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
Let \( ABCD \) be a square. Let \( M \) be the midpoint of \( \overline{DC} \), \( N \) the midpoint of \( \overline{AC} \), and \( P \) the intersection of \( \overline{BM} \) and \( \overline{AC} \). What is the ratio of the area of \( \triangle MNP \) to that of the square \( ABCD \)?

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