
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  $ABCD$  be a rectangle with  $AB=a >CD =b$ . Given circles  $(K_1,r_1) ,  (K_2,r_2)$   with  $r_1<r_2$  tangent externally at point  $K$  and also tangent to the sides of the rectangle, circle  $(K_1,r_1)$  tangent to both  $AD$  and  $AB$ , circle  $(K_2,r_2)$  tangent to both  $AB$  and  $BC$ . Let also the internal common tangent of those circles pass through point  $D$ .
(i) Express sidelengths  $a$  and  $b$  in terms of  $r_1$  and  $r_2$ .
(ii) Calculate the ratios  $\frac{r_1}{r_2}$  and  $\frac{a}{b}$  .
(iii) Find the length of  $DK$  in terms of  $r_1$  and  $r_2$ .

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