
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
For a nonempty set  $\, S \,$  of integers, let  $\, \sigma(S) \,$  be the sum of the elements of  $\, S$ .  Suppose that  $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$  is a set of positive integers with  $\, a_1 < a_2 < \cdots < a_{11} \,$  and that, for each positive integer  $\, n\leq 1500, \,$  there is a subset   $\, S \,$  of  $\, A \,$  for which  $\, \sigma(S) = n$ .  What is the smallest possible value of  $\, a_{10}$ ?

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