
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
Two congruent squares with side length $\sqrt{2}$ share the same center $O$. The length of $\overline{AB}$ is $\frac{1}{2}$. Calculate the area of the octagon $ABCDEFGH$ formed by the lines connecting consecutive vertices of the squares, and express the answer as a reduced fraction $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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