
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 square \(ABCD\), two circles \( \odot O_1 \) and \( \odot O_2 \) are given such that:
- \( \odot O_1 \) and \( \odot O_2 \) are tangent to each other externally,
- \( \odot O_1 \) is tangent to sides \(AB\) and \(AD\),
- \( \odot O_2 \) is tangent to sides \(BC\) and \(CD\).

Given that the side length of square \(ABCD\) is 1, and the radii of \( \odot O_1 \) and \( \odot O_2 \) are \(r_1\) and \(r_2\) respectively:

(i) Find the relationship between \(r_1\) and \(r_2\).

(ii) Find the minimum value of the sum of the areas of \( \odot O_1 \) and \( \odot O_2 \).

For the rectangle \(ABCD\) with width 1 and length \(\frac{3}{2}\), under the same conditions:

Does the sum of the areas of \( \odot O_1 \) and \( \odot O_2 \) have a minimum value? If not, explain why. If it exists, find this minimum value.

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