
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  $ABC$  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  $\frac{c}{a}$ .

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