
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
If the quadratic function $f(x)$ satisfies $f(1+x)=f(1-x)$ for any real number $x$, the minimum value of $f(x)$ is $-1$, and $f(0)=0$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ If the graph of $f(x)$ is always above $y=2x+1+m$ on the interval $[0,1]$, find the range of real number $m$.

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