
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
Points  $A_1,A_2,A_3,...,A_{12}$  are the vertices of a regular polygon in that order. The 12 diagonals  $A_1A_6,A_2A_7,A_3A_8,...,A_{11}A_4,A_{12}A_5$  are marked, as in the picture below. Let  $X$  be some point in the plane. From  $X$ , we draw perpendicular lines to all 12 marked diagonals. Let  $B_1,B_2,B_3,...,B_{12}$  be the feet of the perpendiculars, so that  $B_1$  lies on  $A_1A_6$ ,  $B_2$  lies on  $A_2A_7$  and so on.

Evaluate the ratio  $\frac{XA_1+XA_2+\dots+XA_{12}}{B_1B_6+B_2B_7+\dots+B_{12}B_5}$ .
![Image](https://i.imgur.com/DUuwFth.png)

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