
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
A <u>positive sequence</u> is a finite sequence of positive integers. <u>Sum of a sequence</u> is the sum of all the elements in the sequence. We say that a sequence  $A$  can be <u>embedded</u> into another sequence  $B$ , if there exists a strictly increasing function   $$ \phi : \{1,2, \ldots, |A|\} \rightarrow 
\{1,2, \ldots, |B|\}, $$   such that  $\forall i \in \{1, 2, \ldots ,|A|\}$ ,   $$ A[i] \leq B[\phi(i)], $$   where  $|S|$  denotes the length of
a sequence  $S$ . For example,  $(1,1,2)$  can be embedded in  $(1,2,3)$ , but  $(3,2,1)$  can not be in  $(1,2,3)$ Given a positive integer  $n$ , construct a positive sequence  $U$  with sum  $O(n \, \log \, n)$ , such that all the positive sequences with sum  $n$ , can be embedded into  $U$ .

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