
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 with hypotenuse  $AC$ , and let  $H$  be the foot of the altitude from  $B$  to  $AC$ . Knowing that there is a right-angled triangle with side-lengths  $AB, BC, BH$ , determine all the possible values ​​of  $\frac{AH}{CH}$ 

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