
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
Let  $n$  be a positive integer. Yang the Saltant Sanguivorous Shearling is on the side of a very steep mountain that is embedded in the coordinate plane. There is a blood river along the line  $y=x$ , which Yang may reach but is not permitted to go above (i.e. Yang is allowed to be located at  $(2016,2015)$  and  $(2016,2016)$ , but not at  $(2016,2017)$ ). Yang is currently located at  $(0,0)$  and wishes to reach  $(n,0)$ . Yang is permitted only to make the following moves:

(a) Yang may *spring*, which consists of going from a point  $(x,y)$  to the point  $(x,y+1)$ .
(b) Yang may *stroll*, which consists of going from a point  $(x,y)$  to the point  $(x+1,y)$ .
(c) Yang may *sink*, which consists of going from a point  $(x,y)$  to the point  $(x,y-1)$ .

In addition, whenever Yang does a *sink*, he breaks his tiny little legs and may no longer do a *spring* at any time afterwards. Yang also expends a lot of energy doing a *spring* and gets bloodthirsty, so he must visit the blood river at least once afterwards to quench his bloodthirst. (So Yang may still *spring* while bloodthirsty, but he may not finish his journey while bloodthirsty.) Let there be  $a_n$  different ways for which Yang can reach  $(n,0)$ , given that Yang is permitted to pass by  $(n,0)$  in the middle of his journey. Find the  $2016$ th smallest positive integer  $n$  for which  $a_n\equiv 1\pmod 5$ .

*Proposed by James Lin*

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