
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  $w_{1}$  and  $w_{2}$  be two circles which intersect at points  $A$  and  $B$ . Consider  $w_{3}$  another circle which cuts  $w_{1}$  in  $D,E$ , and it is tangent to  $w_{2}$  in the point  $C$ , and also tangent to  $AB$  in  $F$ . Consider  $G \in DE \cap AB$ , and  $H$  the symetric point of  $F$  w.r.t  $G$ . Find  $\angle{HCF}$ .

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