
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
In the right triangle  $ABC,$   $CF$  is the altitude based on the hypotenuse  $AB.$  
The circle centered at  $B$  and passing through  $F$  and the circle with centre  $A$  and the same radius intersect at a point of  $CB.$  
Determine the ratio  $FB : BC.$ 

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