
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 $\Gamma(S)$ represent the sum of all elements in a non-empty set of integers $S$. Let $A = \{a_1, a_2, \cdots, a_n\}$ be a set of positive integers such that $a_1 < a_2 < \cdots < a_n$. If for each positive integer $n \leq 1500$, there exists a subset $S$ of $A$ such that $\Gamma(S) = n$, find the minimum value of $a_{10}$.

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