
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  $A_1A_2A_3A_4A_5$  be a regular pentagon inscribed in a circle with area  $\tfrac{5+\sqrt{5}}{10}\pi$ . For each  $i=1,2,\dots,5$ , points  $B_i$  and  $C_i$  lie on ray  $\overrightarrow{A_iA_{i+1}}$  such that
\[B_iA_i \cdot B_iA_{i+1} = B_iA_{i+2} \quad \text{and} \quad C_iA_i \cdot C_iA_{i+1} = C_iA_{i+2}^2\]where indices are taken modulo 5. The value of  $\tfrac{[B_1B_2B_3B_4B_5]}{[C_1C_2C_3C_4C_5]}$  (where  $[\mathcal P]$  denotes the area of polygon  $\mathcal P$ ) can be expressed as  $\tfrac{a+b\sqrt{5}}{c}$ , where  $a$ ,  $b$ , and  $c$  are integers, and  $c > 0$  is as small as possible. Find  $100a+10b+c$ .

*Proposed by Robin Park*

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