
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 $f(x)=\log _{2}(x+a)$ where $a \gt 0$. When the point $P(x,y)$ moves on the graph of the function $y=g(x)$, the corresponding point $P_{0}(4x,2y)$ moves on the graph of the function $y=f(x)$, then the function $y=g(x)$ is called the "companion" function of the function $y=f(x)$.
- $(Ⅰ)$ Solve the inequality with respect to $x$: $f(x) \lt -1$.
- $(Ⅱ)$ For any $x\in (0,2)$, the graph of $f(x)$ is always below the graph of its "companion" function, find the range of $a$.
- $(Ⅲ)$ Let $F(x)=f(x)-g(x)$, $x\in (0,2)$. When $a=1$, find the maximum value of $|F(x)|$.

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