
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 $f\left(x\right)=\log _{2}(x+a)$.  
$(1)$ When $a=2$, solve the inequality: $f\left(x\right) \lt 2\log _{2}x$;  
$(2)$ If the maximum value of the function $y=|f\left(x\right)|$ on $x\in \left[-1,2\right]$ is $\log _{2}3$, find the value of $a$;  
$(3)$ When $a \gt 0$, let $g(x)=\frac{1}{2}f(4x)$. If for any $x\in \left(0,2\right)$, the graph of the function $y=f\left(x\right)$ is always below the graph of the function $y=g\left(x\right)$, find the range of positive values of $a$.

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