
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 triangle \(ABC\), angle \(\hat{B}\) is \(\pi / 4\) and angle \(\hat{C}\) is \(\pi / 6\). Circles are constructed on the medians \(BM\) and \(CN\), using them as diameters. These circles intersect at points \(P\) and \(Q\). The chord \(PQ\) intersects side \(BC\) at point \(D\). Find the ratio \(|BC| : |DC|\.

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