
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
A certain type of sapling has a height of $A$ meters when planted, where $A$ is a constant. The height of the sapling $n$ years after planting is denoted as $f(n)$. Research has found that $f(n)$ approximately satisfies $f(n)= \dfrac{9A}{a+bt^{n}}$, where $t=2^{- \frac{2}{3}}$, $a$ and $b$ are constants, and $n\in\mathbb{N}$, $f(0)=A$. It is known that the height of the tree is three times its initial height after being planted for 3 years.

$(1)$ After how many years of planting will the height of the tree be eight times its initial height?

$(2)$ In which year after planting will the tree have its maximum growth in height?

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