
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
Using $\sigma(\mathrm{s})$ to represent the sum of all elements in a non-empty set of integers $\mathrm{S}$, let $A = \{a_{1}, a_{2}, \cdots, a_{11}\}$ be 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 $\mathrm{S}$ of $\mathrm{A}$ such that $\sigma(S) = n$, then find the minimum value of $a_{10}$ satisfying the above requirements.

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