
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]


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Let  $S$  be the set of the reciprocals of the first  $2016$  positive integers and  $T$  the set of all subsets of  $S$  that form arithmetic progressions. What is the largest possible number of terms in a member of  $T$ ?

*2016 CCA Math Bonanza Lightning #3.4*

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