
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]


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Define: If there exists a constant \( k \) such that for any two different real numbers \( x_1 \) and \( x_2 \) within the domain \( D \), the inequality \( \left| f(x_1) - f(x_2) \right| \leq k \left| x_1 - x_2 \right| \) holds, then the function \( f(x) \) is said to satisfy the Lipschitz condition on \( D \).
(1) Give an example of a function that satisfies the Lipschitz condition and state the value of the constant \( k \), and verify the condition;
(2) If the function \( f(x) = \sqrt{x+1} \) satisfies the Lipschitz condition on \( [0, +\infty) \), find the minimum value of the constant \( k \).

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