
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 $f(x)$ such that $f(0)=0$, please choose one of the following conditions ① and ② as the known condition to complete the following questions. <br/> ① $f(x+2)=f(x+1)+2x+1$; ② The solution set of the inequality $f(x) < x+4$ is $\left(-1,4\right)$. <br/> $(1)$ Find the analytical expression of $f(x)$; <br/> $(2)$ If the range of $f(x)$ on $\left[-1,m\right]$ is $\left[-1,3\right]$, find the range of real number $m$.

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