
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
(a) Determine all real solutions to \( a^2 + 10 = a + 10^2 \).

(b) Determine two positive real numbers \( a, b > 0 \) such that \( a \ne b \) and \( a^2 + b = b^2 + a \).

(c) Find all triples of real numbers \( (a, b, c) \) such that \( a^2 + b^2 + c = b^2 + c^2 + a = c^2 + a^2 + b \).

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