
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]


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Let  $n\geq 2$  be a fixed positive integer. Let  $\{a_1,a_2,...,a_n\}$  be fixed positive integers whose sum is  $2n-1$ . Denote by  $S_{\mathbb{A}}$  the sum of elements of a set  $A$ . Find the minimal and maximal value of  $S_{\mathbb{X}}\cdot S_{\mathbb{Y}}$  where  $\mathbb{X}$  and  $\mathbb{Y}$  are two sets with the property that  $\mathbb{X}\cup \mathbb{Y}=\{a_1,a_2,...,a_n\}$  and  $\mathbb{X}\cap \mathbb{Y}=\emptyset.$ *Note:  $\mathbb{X}$  and  $\mathbb{Y}$  can have multiple equal elements. For example, when  $n=5$  and  $a_1=...=a_4=1$  and  $a_5=5$ , we can consider  $\mathbb{X}=\{1,1,1\}$  and  $\mathbb{Y}=\{1,5\}$ . Moreover, in this case,  $S_\mathbb{X}=3$  and  $S_{\mathbb{Y}}=6.$*

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