
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
From point $A$ on plane, there are rays $e_{0}, e_{1}, \ldots, e_{6}$ emanating from it, where $e_{1}$ forms a $30^{\circ}$ angle with $e_{0}$, and each subsequent ray forms a $30^{\circ}$ angle with the previous one. Project the point $B_{0}$, which is at a unit distance from $e_{0} A$, perpendicularly onto $e_{1}$, and let this projection be $B_{1}$. Similarly, project $B_{1}$ onto $e_{2}$ to get $B_{2}$ and so on. Calculate the length of the broken line $B_{0} B_{1} \ldots B_{6}$, and the area of the polygon $B_{0} B_{1} \ldots B_{6}$.

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