
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 $A = \sqrt{2} - 1$, $B = \sqrt{3} - \sqrt{2}$, $C = \sqrt{4} - \sqrt{3}$:
(I) Compare the sizes of $A$ and $B$, as well as $B$ and $C$ separately (only state the results, no need to provide the proof);
(II) Based on the results from (I), conjecture the relationship between $\sqrt{k} - \sqrt{k-1}$ and $\sqrt{k+1} - \sqrt{k}$ for $k \geq 2$ and $k \in \mathbb{N}^*$, and provide a proof.

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