
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
Let $f(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$, with $(a, b)$ and $(b, a)$ being distinct if $a \neq b$. What is the smallest positive integer $n$ for which $f(n) = 4$?

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