
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
22. Let the function  $f : \mathbb{Z} \to \mathbb{Z}$  take only integer inputs and have integer outputs. For any integers  $x$  and  $y$ , f satisfies  $$ f(x) + f(y) = f(x + 1) + f(y - 1) $$  If  $f(2016) = 6102$  and  $f(6102) = 2016$ , what is  $f(1)$ ?

23. Let  $d$  be a randomly chosen divisor of  $2016$ . Find the expected value of  $$ \frac{d^2}{d^2 + 2016} $$  24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of  $N$  colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of  $N$ ?

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