
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
**i.)** Consider a circle  $K$  with diameter  $AB;$  with circle  $L$  tangent to  $AB$  and to  $K$  and with a circle  $M$  tangent to circle  $K,$  circle  $L$  and  $AB.$  Calculate the ration of the area of circle  $K$  to the area of circle  $M.$ 

**ii.)** In triangle  $ABC, AB = AC$  and  $\angle CAB = 80^{\circ}.$  If points  $D,E$  and  $F$  lie on sides  $BC, AC$  and  $AB,$  respectively and  $CE = CD$  and  $BF = BD,$  then find the size of  $\angle EDF.$ 

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