
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 the function $f$ be defined on the set of ordered pairs of positive integers, and satisfy:
1. $f(x, x) = x$;
2. $f(x, y) = f(y, x)$;
3. $(x+y)f(x, y) = yf(x, x+y)$. Calculate the value of $f(12, 16) + f(16, 12)$.

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