
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
Every October in the golden autumn, the off-road $e$ tribe's Alashan Heroes Meeting is held in Tengger Desert, Alxa League, Inner Mongolia Autonomous Region. The event has been developed into a super car culture event entertainment complex integrating desert sports, ultimate car culture experience, and themed leisure vacation. In order to reduce environmental pollution, a certain environmental protection department rented special environmental protection vehicles to clean up the garbage on site. By consulting the attendance numbers $x$ (in ten thousand people) and the number of environmental protection vehicles required in the desert $y$ (vehicles) for the past five years, the following statistical table was obtained:

| Attendance $x$ (ten thousand people) | 11 | 9 | 8 | 10 | 12 |
|--------------------------------------|----|---|---|----|----|
| Number of environmental protection vehicles $y$ (vehicles) | 28 | 23 | 20 | 25 | 29 |

$(Ⅰ)$ Based on the five sets of data given in the statistical table, find the linear regression equation of $y$ with respect to $x$, $\hat{y} = \hat{b}x + \hat{a}$.

$(Ⅱ)$ It is known that the relationship between the average cost $C$ per rented environmental protection vehicle and the quantity $t$ (vehicles) is $C = \left\{\begin{array}{l}{3000t + 200, 0 < t < 35, t \in \mathbb{N}}\\{2900t, t \geq 35, t \in \mathbb{N}}\end{array}\right.$. According to the actual number of attendees, the organizer needs to invest in the environmental protection vehicles. Each vehicle costs $6000$ yuan, and the organizer will not pay for any vehicles beyond the actual requirement. It is estimated that about $140,000$ people will participate in this Heroes Meeting. Based on the linear regression equation obtained in $(Ⅰ)$, predict how many environmental protection vehicles the environmental protection department should rent to ensure the completion of the cleaning task. What is the profit $L$ obtained?

(Note: Profit $L = $ amount paid by the organizer $-$ cost of rented vehicles). 

Reference formula: $\hat{b} = \frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}} = \frac{\sum_{i=1}^{n}x_{i}y_{i}-n\overline{xy}}{\sum_{i=1}^{n}x_{i}^{2}-n\overline{x}^{2}}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$.

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