
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
Beyond the Point of No Return is a large lake containing 2013 islands arranged at the vertices of a regular  $2013$ -gon.  Adjacent islands are joined with exactly two bridges.  Christine starts on one of the islands with the intention of burning all the bridges.  Each minute, if the island she is on has at least one bridge still joined to it, she randomly selects one such bridge, crosses it, and immediately burns it. Otherwise, she stops.

If the probability Christine burns all the bridges before she stops can be written as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , find the remainder when  $m+n$  is divided by  $1000$ .

*Evan Chen*

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