
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

...

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system
Chinese tea culture is profound, and the taste of tea is related to the type of tea leaves and the temperature of the water. Water City Chun Tea is renowned both inside and outside Guizhou Province for its rich content of organic tea selenium and more than ten essential trace elements for the human body. Experience shows that Water City Chun Tea is best brewed with water at $85^{\circ}\mathrm{C}$, and the optimal taste is achieved when the tea water temperature drops to $60^{\circ}\mathrm{C}$. To facilitate temperature control, a research group used a cooling model for the temperature change of an object in a room temperature environment: if the initial temperature of the object is $\theta _{1}^{\circ}\mathrm{C}$, and the room temperature is $\theta _{0}^{\circ}\mathrm{C}$, then after time $t$ (in minutes), the temperature $\theta$ of the object satisfies $\theta = {\theta}_{0} + ({\theta}_{1} - {\theta}_{0}) \cdot e^{-kt}$, where $k$ is a positive constant. Under room temperature of $19^{\circ}\mathrm{C}$, the research group, through multiple measurements and averaging, obtained the time required for the temperature of $200mL$ of water at an initial temperature of $98^{\circ}\mathrm{C}$ to drop to the corresponding temperatures as shown in the table below:

| Time to drop from $98^{\circ}\mathrm{C}$ to $85^{\circ}\mathrm{C}$ | 3.4 minutes |
|--------------------------------------------------------------------|--------------|
| Time to drop from $98^{\circ}\mathrm{C}$ to $80^{\circ}\mathrm{C}$ | 5.0 minutes |

$(1)$ Select a set of data from the table above to find the value of $k$ (accurate to $0.01$), and write the analytical expression of the cooling time $t$ with respect to the cooled water temperature $\theta$ based on the cooling model mentioned above.  
$(2)$ Under the conditions of $(1)$, now brew Water City Chun Tea with $200mL$ of water at a room temperature of $19^{\circ}\mathrm{C}$. How many minutes are needed from brewing to achieve the optimal taste for drinking? (Round to $0.1$ minutes)  
(Reference data: $\ln 79 \approx 4.369, \ln 66 \approx 4.190, \ln 61 \approx 4.111, \ln 41 \approx 3.714$)

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