
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
Given two spherical balls of different sizes placed in two corners of a rectangular room, where each ball touches two walls and the floor, and there is a point on each ball such that the distance from the two walls it touches to that point is 5 inches and the distance from the floor to that point is 10 inches, find the sum of the diameters of the two balls.

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