
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 a rectangular prism \( ABCD-A_1B_1C_1D_1 \), where \( AB = 5 \), \( AD = 3 \), and \( AA_1 = 4 \), a plane \(\alpha\) passing through point \( A \) and parallel to line \( CD \) divides the prism into two parts. Now, two spheres are placed inside these two parts of the prism. As the plane \(\alpha\) changes, what is the maximum value of the sum of the radii of these two spheres?

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