
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
Given a parallelogram \(ABCD\) where \(AB = a\) and \(AD = b\). The first circle has its center at vertex \(A\) and passes through \(D\), and the second circle has its center at \(C\) and passes through \(D\). An arbitrary circle with center \(B\) intersects the first circle at points \(M_1, N_1\) and the second circle at points \(M_2, N_2\). What is the ratio \(M_1 N_1 : M_2 N_2\)?

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