
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  $k$  distinct integers  $a_1,a_2,...,a_k$  such that there exists an injective function  $f$  from reals to themselves such that for each positive integer  $n$  we have  $$ \{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\} $$  .

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