
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  $S=\{1,2, \ldots ,10^{10}\}$ . Find all functions  $f:S \rightarrow S$ , such that   $$ f(x+1)=f(f(x))+1 \pmod {10^{10}} $$   for each  $x \in S$  (assume  $f(10^{10}+1)=f(1)$ ).

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