
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 that \(ABCD\) is a square, points \(E\) and \(F\) lie on the side \(BC\) and \(CD\) respectively, such that \(BE = CF = \frac{1}{3} AB\). \(G\) is the intersection of \(BF\) and \(DE\). If
\[
\frac{\text{Area of } ABGD}{\text{Area of } ABCD} = \frac{m}{n}
\]
is in its lowest terms, find the value of \(m+n\).

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