
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 the function $f(x)=1-\frac{2}{2^{x}+1}$.  
$(1)$ Determine the monotonicity of the function and provide a proof.  
$(2)$ If the equation $f(x)=m$ has solutions in the interval $\left[-2,2\right]$, find the range of real numbers for $m$.

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