
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 $f(x)$ that satisfies $f(x+1)-f(x)=2x-1$, and $f(0)=3$.
(1) Find the analytic expression of $f(x)$;
(2) If for any distinct $x\_1$, $x\_2∈(2,4)$, $|f(x\_1)-f(x\_2)| < k|x\_1-x\_2|$ holds, find the range of real number $k$.

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