
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
Arianna and Brianna play a game in which they alternate turns writing numbers on a paper. Before the game begins, a referee randomly selects an integer  $N$  with  $1 \leq N \leq 2019$ , such that  $i$  has probability  $\frac{i}{1 + 2 + \dots + 2019}$  of being chosen. First, Arianna writes  $1$  on the paper. On any move thereafter, the player whose turn it is writes  $a+1$  or  $2a$ , where  $a$  is any number on the paper, under the conditions that no number is ever written twice and any number written does not exceed  $N$ . No number is ever erased. The winner is the person who first writes the number  $N$ . Assuming both Arianna and Brianna play optimally, the probability that Brianna wins can be expressed as  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Compute  $m + n.$ *Proposed by Edward Wan*

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