
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 quadratic function $f(x)$ satisfying $f(x+1) - f(x) = 2x - 1$, and $f(0) = 3$.
(1) Find the analytical expression for $f(x)$.
(2) If for any pair of distinct $x_1, x_2 \in (2, 4)$, the inequality $|f(x_1) - f(x_2)| < k|x_1 - x_2|$ holds true, determine the range of the real number $k$.

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