
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
Three points  $A,B,C$  are such that  $B \in ]AC[$ . On the side of  $AC$  we draw the three semicircles with diameters  $[AB], [BC]$  and  $[AC]$ . The common interior tangent at  $B$  to the first two semi-circles meets the third circle in  $E$ . Let  $U$  and  $V$  be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle  $ABC$  as  $S(ABC)$ . Evaluate the ratio  $R=\frac{S(EUV)}{S(EAC)}$  as a function of  $r_1 = \frac{AB}{2}$  and  $r_2 = \frac{BC}{2}$ .

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