
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  $S$  be a subset of  $\{0,1,2,\dots ,9\}$ . Suppose there is a positive integer  $N$  such that for any integer  $n>N$ , one can find positive integers  $a,b$  so that  $n=a+b$  and all the digits in the decimal representations of  $a,b$  (expressed without leading zeros) are in  $S$ . Find the smallest possible value of  $|S|$ .

*Proposed by Sutanay Bhattacharya*

<details><summary>Original Wording</summary>As pointed out by Wizard_32, the original wording is:

Let  $X=\{0,1,2,\dots,9\}.$  Let  $S \subset X$  be such that any positive integer  $n$  can be written as  $p+q$  where the non-negative integers  $p, q$  have all their digits in  $S.$  Find the smallest possible number of elements in  $S.$</details>

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