
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
In the country of Graphia there are  $100$  towns, each numbered from  $1$  to  $100$ . Some pairs of towns may be connected by a (direct) road and we call such pairs of towns  *adjacent*. No two roads connect the same pair of towns.

Peter, a foreign tourist, plans to visit Graphia  $100$  times. For each  $i$ ,  $i=1,2,\dots, 100$ , Peter starts his  $i$ -th trip by arriving in the town numbered  $i$  and then each following day Peter travels from the town he is currently in to an adjacent town with the lowest assigned number, assuming such that a town exists and that he hasn't visited it already on the  $i$ -th trip. Otherwise, Peter deems his  $i$ -th trip to be complete and returns home. 

It turns out that after all  $100$  trips, Peter has visited each town in Graphia the same number of times. Find the largest possible number of roads in Graphia.

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