
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
A particle moves from  $(0, 0)$  to  $(n, n)$  directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At  $(n, y), y < n$ , it stays there if a head comes up and at  $(x, n), x < n$ , it stays there if a tail comes up. Let $ k$  be a fixed positive integer. Find the probability that the particle needs exactly  $2n+k$  tosses to reach  $(n, n).$ 

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