
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
To the numbers 5, 7, and 10, the same number is added such that the squares of the results form three consecutive terms of an arithmetic sequence. What is the common difference of the sequence, and what number was added to the original numbers?

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