
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
The acute angle of a right-angled triangle is \(\alpha\). Find the ratio of the radius of the inscribed circle to the radius of the circumscribed circle. For which value of \(\alpha\) is this ratio the greatest?

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