<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Let  $ABC$  be a triangle with  $\angle ABC=90^{\circ}$ . The square  $BDEF$  is inscribed in  $\triangle ABC$ , such that  $D,E,F$  are in the sides  $AB,CA,BC$  respectively. The inradius of  $\triangle EFC$  and  $\triangle EDA$  are  $c$  and  $b$ , respectively. Four circles  $\omega_1,\omega_2,\omega_3,\omega_4$  are drawn inside the square  $BDEF$ , such that the radius of  $\omega_1$  and  $\omega_3$  are both equal to  $b$  and the radius of  $\omega_2$  and  $\omega_4$  are both equal to  $a$ . The circle  $\omega_1$  is tangent to  $ED$ , the circle  $\omega_3$  is tangent to  $BF$ ,  $\omega_2$  is tangent to  $EF$  and  $\omega_4$  is tangent to  $BD$ , each one of these circles are tangent to the two closest circles and the circles  $\omega_1$  and  $\omega_3$  are tangents. Determine the ratio  $\frac{c}{a}$ . Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
2
Let  $ABC$  be a triangle with  $\angle ABC=90^{\circ}$ . The square  $BDEF$  is inscribed in  $\triangle ABC$ , such that  $D,E,F$  are in the sides  $AB,CA,BC$  respectively. The inradius of  $\triangle EFC$  and  $\triangle EDA$  are  $c$  and  $b$ , respectively. Four circles  $\omega_1,\omega_2,\omega_3,\omega_4$  are drawn inside the square  $BDEF$ , such that the radius of  $\omega_1$  and  $\omega_3$  are both equal to  $b$  and the radius of  $\omega_2$  and  $\omega_4$  are both equal to  $a$ . The circle  $\omega_1$  is tangent to  $ED$ , the circle  $\omega_3$  is tangent to  $BF$ ,  $\omega_2$  is tangent to  $EF$  and  $\omega_4$  is tangent to  $BD$ , each one of these circles are tangent to the two closest circles and the circles  $\omega_1$  and  $\omega_3$  are tangents. Determine the ratio  $\frac{c}{a}$ .
test