
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 $f(x)=3^{x}$ with domain $\mathbb{R}$, and it satisfies $f(a+2)=18$, the function $g(x)=\lambda \cdot 3^{ax}-4^{x}$ has the domain $[0,1]$.  
$(1)$ Find the value of the real number $a$;  
$(2)$ If the function $g(x)$ is a monotonically decreasing function on its domain, find the range of the real number $\lambda$;  
$(3)$ When $\lambda$ takes what value, the maximum value of the function $g(x)$ is $\frac{1}{2}$.

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