
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
Given a set of positive integers 
$$
S=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}
$$

such that for any subsets $S_{1}$ and $S_{2}$ of $S$, where $S_{1} \neq S_{2}$, the condition holds:
$$
\sum_{i \in S_{1}} i \neq \sum_{i \in S_{2}} j.
$$

Determine the minimum value of $\sqrt{a_{1}}+\sqrt{a_{2}}+\cdots+\sqrt{a_{n}}$.

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