
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 $S$ be a non-empty set composed of integers, and let $\sigma(S)$ represent the sum of all elements in $S$. Consider $A=\left\{a_{1}, a_{2}, \cdots, a_{11}\right\}$ to be a set of positive integers such that $a_{1}<a_{2}<\cdots<a_{11}$. If for every positive integer $n \leqslant 1500$, there exists a subset $S$ of $A$ such that $\sigma(S)=n$, determine the smallest possible value of $a_{10}$.

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