
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 $f(x) = 3^x$, it is known that $f(a+2) = 27$, and the domain of the function $g(x) = \lambda \cdot 2^{ax} - 4^x$ is $[0, 2]$.
(1) Find the value of $a$.
(2) If the function $g(x)$ is monotonically decreasing on $[0, 2]$, determine the range of $\lambda$.
(3) If the maximum value of the function $g(x)$ is $1$, find the value of $\lambda$.

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