
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
An approximate method is known for constructing a regular heptagon inscribed in a circle of radius $r$, where instead of using the side of the heptagon, the height $m$ of an equilateral triangle of side $r$ is used. Knowing that the side of a regular heptagon to six decimal places is $0.867768r$, calculate $m$ to six decimal places as well, then determine what percentage of the exact length of the side is the difference between these two values; this can be considered the measure of accuracy, or error, of the given approximation method.

Is the following approximate method better? Let $A$ and $B$ be points on the circle that are $r$ apart, measure a segment seven times along the line $AB$ from $A$ towards $B$, connect the mirror image $D$ of the point $C$ (the endpoint of the seventh segment) relative to the center $O$ of the circle to the endpoint $E$ of the second segment, finally label the intersection point $F$ of $DE$ and the circle's radius $OA$ and consider $BF$ as the approximate length of the heptagon’s side.

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