
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 $f(x+1)=a\log _{3}(x+1)$, and $f(2)=1$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ Given that the domain of $f(x)$ is $\left[2,+\infty \right)$. If the equation $f(4^{x}+1)-f(k\cdot 2^{x}+k)=x$ has a unique real root, find the range of real number $k$.

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