
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 $f(x) = ax^2 + bx + c$, satisfying $f(0) = 2$, and $f(x+1) - f(x) = 2x - 1$.  
(Ⅰ) Find the expression of the function $f(x)$.  
(Ⅱ) If the inequality $f(x) - t > 0$ has solutions on the interval $[-1, 2]$, find the range of the real number $t$.  
(Ⅲ) If the function $g(x) = f(x) - mx$ has two zeros, one in the interval $(-1, 2)$ and the other in the interval $(2, 4)$, find the range of the real number $m$.

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