
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
Given a number  $n\in\mathbb{Z}^+$  and let  $S$  denotes the set  $\{0,1,2,...,2n+1\}$ . Consider the function  $f:\mathbb{Z}\times S\to [0,1]$  satisfying two following conditions simultaneously:

i)  $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$ ;
ii)  $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$  for all  $x\in\mathbb{Z}$  and  $y\in\{1,2,3,...,2n\}$ .

Let  $F$  be the set of such functions. For each  $f\in F$ , let  $v(f)$  be the set of values of  $f$ .

a) Proof that  $|F|=\infty$ .

b) Proof that for each  $f\in F$  then  $|v(f)|<\infty$ .

c) Find the maximum value of  $|v(f)|$  for  $f\in F$ .

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