
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
Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin  $2013$  times. On the  $n^{\text{th}}$  flip (where  $n=1,2,\dots,2013$ ), Ben does the following if the coin flips heads:
  (i) If the blackboard is empty, Ben writes  $n$  on the blackboard.
  (ii) If the blackboard is not empty, let  $m$  denote the largest number on the blackboard.  If  $m^2+2n^2$  is divisible by  $3$ , Ben erases  $m$  from the blackboard; otherwise, he writes the number  $n$ .
No action is taken when the coin flips tails. If probability that the blackboard is empty after all  $2013$  flips is  $\frac{2u+1}{2^k(2v+1)}$ , where  $u$ ,  $v$ , and  $k$  are nonnegative integers, compute  $k$ .

*Proposed by Evan Chen*

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