
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
A square has sides of length $a$. Opposite sides are each divided into $n^2$ congruent segments, with the endpoints of one of these segments joined to the center of the square forming triangle $A$. The other sides are each divided into $m^2$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. Calculate the ratio of the area of triangle $A$ to the area of triangle $B$.
A) $\frac{1}{n^2 m^2}$
B) $\frac{n^2}{m^2}$
C) $\frac{m^2}{n^2}$
D) $n^2 m^2$
E) $\frac{1}{2}(n^2 + m^2)$

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