
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
Find all surjective functions  $f:\mathbb{N}\to\mathbb{N}$  such that for all positive integers  $a$  and  $b$ , exactly one of the following equations is true:
\begin{align*}
f(a)&=f(b), <br /> 
 f(a+b)&=\min\{f(a),f(b)\}.
\end{align*}
*Remarks:*  $\mathbb{N}$  denotes the set of all positive integers. A function  $f:X\to Y$  is said to be surjective if for every  $y\in Y$  there exists  $x\in X$  such that  $f(x)=y$ .

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