
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 $f(x)$ satisfies $f(x+1)-f(x)=2x$ (where $x \in \mathbb{R}$), and $f(0)=1$.
1. Find the expression for $f(x)$.
2. If the function $g(x)=f(x)-2tx$ is monotonic on the interval $[-1, 5]$, find the range of the real number $t$.
3. If the equation $f(x)=x+m$ has a unique real root in the interval $(-1, 2)$, find the range of the real number $m$ (Note: identical real roots are counted as one).

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