
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
On the diagonal \( BD \) of parallelogram \( ABCD \), points \( P \) and \( Q \) are chosen such that \( BP = PQ = QD \). Lines \( CP \) and \( CQ \) intersect sides \( AB \) and \( AD \) at points \( M \) and \( N \) respectively. Find the ratio \( MN : BD \).

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