
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 right-angled triangle ( $\angle C = 90^\circ$ ) and  $D$  be the midpoint of an altitude from C. The reflections of the line  $AB$  about  $AD$  and  $BD$ , respectively, meet at point  $F$ . Find the ratio  $S_{ABF}:S_{ABC}$ . 
Note:  $S_{\alpha}$  means the area of  $\alpha$ . 

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