
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 that $x, y \in (0,+\infty)$, and $x^{2}+y^{2}=x+y$.

(I) Find the minimum value of $\frac{1}{x}+\frac{1}{y}$;

(II) Does there exist $x$ and $y$ such that $(x+1)(y+1)=5$? Please explain your answer.

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