
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 one side of  $AC$ , draw the three semicircles with diameters  $AB,BC,CA$ . The common interior tangent at  $B$  to the first two semicircles meets the third circle  $E$ . Let  $U,V$  be the points of contact of the common exterior tangent to the first two semicircles.
Evaluate the ratio  $R=\frac{[EUV]}{[EAC]}$  as a function of  $r_{1} = \frac{AB}{2}$  and  $r_2 = \frac{BC}{2}$ , where  $[X]$  denotes the area of polygon  $X$ .

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