
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
Consider a finite set $T$ of positive integers. When the greatest integer in $T$ is removed, the average of the remaining integers becomes $34$. If the least integer in $T$ is also removed, the average then becomes $37$. When the greatest integer is reintroduced to the set, the average becomes $42$. The greatest integer in the original set $T$ is $80$ greater than the least integer. Find the average value of all integers in the set $T$.
A) 37.8 
B) 38.2 
C) 38.82 
D) 39.2 
E) 40.0

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