
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  $n_1,n_2, \cdots, n_{26}$  be pairwise distinct positive integers satisfying
(1) for each  $n_i$ , its digits belong to the set  $\{1,2\}$ ;
(2) for each  $i,j$ ,  $n_i$  can't be obtained from  $n_j$  by adding some digits on the right.
Find the smallest possible value of  $\sum_{i=1}^{26} S(n_i)$ , where  $S(m)$  denotes the sum of all digits of a positive integer  $m$ .

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