
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
Adam, Bendeguz, Cathy, and Dennis all see a positive integer  $n$ . Adam says, " $n$  leaves a remainder of  $2$  when divided by  $3$ ." Bendeguz says, "For some  $k$ ,  $n$  is the sum of the first  $k$  positive integers." Cathy says, "Let  $s$  be the largest perfect square that is less than  $2n$ . Then  $2n - s = 20$ ." Dennis says, "For some  $m$ , if I have  $m$  marbles, there are  $n$  ways to choose two of them." If exactly one of them is lying, what is  $n$ ?

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