
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  $ ABCDEF$  be a regular hexagon. Let  $ G$ ,  $ H$ ,  $ I$ ,  $ J$ ,  $ K$ , and  $ L$  be the midpoints of sides  $ AB$ ,  $ BC$ ,  $ CD$ ,  $ DE$ ,  $ EF$ , and  $ AF$ , respectively. The segments  $ AH$ ,  $ BI$ ,  $ CJ$ ,  $ DK$ ,  $ EL$ , and  $ FG$  bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of  $ ABCDEF$  be expressed as a fraction  $ \frac {m}{n}$  where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m \plus{} n$ .

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