
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
Use \(\sigma(S)\) to represent the sum of all elements in a non-empty set of integers \(S\). Suppose \(A=\{a_{1}, a_{2}, \ldots, a_{11}\}\) is a set of positive integers such that \(a_{1}<a_{2}<\cdots<a_{11}\). If for every positive integer \(n \leq 1500\), there exists a subset \(S\) of \(A\) such that \(\sigma(S)=n\), determine the minimum value of \(a_{10}\) which satisfies the above condition.

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