
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 function $f(x)$ with domain $\mathbb{R}$ and satisfying:
1. $f(1) = 1 > f(-1)$;
2. For any real numbers $x$ and $y$, $f(y-x+1) = f(x)f(y) + f(x-1)f(y-1)$.

(I) Find the values of $f(0)$ and $f(3)$.
(II) Determine the odd-even symmetry and periodicity of the function and find the value of $f^2(3x) + f^2(3x-1)$.
(III) Determine whether there exist constants $A$ and $B$ such that the inequality $|f(x) + f(2-x) + Ax + B| \leq 2$ holds for all real numbers $x$. If such constants exist, find the values of $A$ and $B$; if not, explain why.

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