
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 $f(0)=0$ and $f(x+1)=f(x)+x+1$,

(1) Find the expression for $f(x)$.

(2) Find the minimum value $g(t)$ of the function $f(x)$ on the interval $[t, t+1]$.

(3) If $g(t)+m \geq 0$ holds for all $t \in \mathbb{R}$, find the range of the real number $m$.

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