
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
Given two positive integers  $m,n$ , we say that a function  $f : [0,m] \to \mathbb{R}$  is  $(m,n)$ -*slippery* if it has the following properties:

i)  $f$  is continuous;
ii)  $f(0) = 0$ ,  $f(m) = n$ ;
iii) If  $t_1, t_2\in [0,m]$  with  $t_1 < t_2$  are such that  $t_2-t_1\in \mathbb{Z}$  and  $f(t_2)-f(t_1)\in\mathbb{Z}$ , then  $t_2-t_1 \in \{0,m\}$ .

Find all the possible values for  $m, n$  such that there is a function  $f$  that is  $(m,n)$ -slippery.

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