
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
There are many sets of two different positive integers \( a \) and \( b \), both less than 50, such that \( a^2 \) and \( b^2 \) end in the same last two digits. For example, \( 35^2 = 1225 \) and \( 45^2 = 2025 \) both end in 25. What are all possible values for the average of \( a \) and \( b \)?

For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider \( 2^2 \) to end with the digits 04, not 4.

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