
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
Due to insufficient anticipation of the negative impact of highly developed industry and inadequate prevention, three major global crises have emerged: resource scarcity, environmental pollution, and ecological destruction. Environmental pollution refers to the addition of certain substances to the environment, either naturally or artificially, exceeding the environment's self-purification capacity and causing harm (or due to human factors, the environment is contaminated by harmful substances, affecting the growth, reproduction of organisms, and the normal life of humans). Changes in the composition or state of the environment caused by human factors lead to a decrease in environmental quality, disrupting and destroying the normal production and living conditions of ecosystems and humans. According to research, a certain pollutant has a very strong pollution power. Investigation and research are currently being conducted on the pollution power of this pollutant. Through experimental investigations, data on the pollution time $t$ (hours) after the arrival of this pollutant and the pollution area $y$ (square meters) of this pollutant in a certain region can be obtained as follows:

| $t$   | 1      | 2        | 3        | 4        |
|-------|--------|----------|----------|----------|
| $y$   | $e$    | $e^{3}$  | $e^{4}$  | $e^{5}$  |

Through analysis, it is known that there is a strong linear regression relationship between the data $t$ and $\ln y$.

$(1)$ Find the relationship between $t$ and $y$.

$(2)$ Based on the relationship in $(1)$, at what time after the arrival of this pollutant does the pollutant area grow at the slowest rate?

Reference formula: For a set of data $(u_{i}, v_{i})$ $(i=1,2,3,\cdots,n)$, the least squares estimates of the slope and intercept of the regression line $\hat{v}=\hat{\beta}u+\hat{a}$ are respectively $\hat{\beta}=\frac{{\sum_{i=1}^{n}{u_{i}v_{i}}-n\overline{u}\overline{v}}}{{\sum_{i=1}^{n}u_{i}^{2}-n\overline{u}^{2}}}$, $\hat{a}=\overline{v}-\hat{\beta}\overline{u}$.

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