
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
Let $ABC$ be an equilateral triangle. Let $D$, $E$, and $F$ be the midpoints of sides $AB$, $BC$, and $CA$, respectively. The segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ intersect at points $G$, $H$, and $I$ forming a smaller triangle within $ABC$. Calculate the ratio of the area of triangle $GHI$ to triangle $ABC$ and express it as a fraction $\frac {m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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