
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 $a \gt 0$, $b \gt 0$, and $a+b=2$.
- $(1)$ Find the maximum value of $ab$.
- $(2)$ Find the minimum value of $\frac{1}{a}+\frac{1}{b}$.
- $(3)$ If the inequality $|x+m|-|x+1|≤\frac{1}{a}+\frac{1}{b}$ holds for any $x\in \mathbb{R}$ and for any $a$ and $b$ satisfying the given conditions, find the range of real numbers for $m$.

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