
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
Assume  $A=\{a_{1},a_{2},...,a_{12}\}$  is a set of positive integers such that for each positive integer  $n \leq 2500$   there is a subset  $S$  of  $A$  whose sum of elements is  $n$ . If  $a_{1}<a_{2}<...<a_{12}$  , what is the smallest possible value of  $a_{1}$ ?

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