
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
Carbon neutrality refers to the total amount of carbon dioxide or greenhouse gas emissions directly or indirectly generated by a country, enterprise, product, activity, or individual within a certain period of time. It aims to offset the carbon dioxide or greenhouse gas emissions generated by itself through tree planting, energy conservation, and other forms to achieve a relative "zero emissions." On September 22, 2020, the Chinese government proposed at the 75th session of the United Nations General Assembly: "China will increase its nationally determined contributions, adopt more effective policies and measures, strive to peak carbon dioxide emissions before 2030, and endeavor to achieve carbon neutrality by 2060." A factory responded to the national call, continuously upgrading new technologies for treating industrial waste gases, resulting in a monthly decrease in carbon dioxide emissions over the past six months. The specific data is as follows:

| Month Number $(t_{i})$ | 1 | 2 | 3 | 4 | 5 | 6 |
|------------------------|---|---|---|---|---|---|
| Carbon Emission $p_{i}$ (tons) | 100 | 70 | 50 | 35 | 25 | 20 |

Given that $\sum_{i=1}^{6}{{t}_{i}^{2}}=91$, $\sum_{i=1}^{6}{{t}_{i}}ln{p}_{i}≈73.1$, $\sum_{i=1}^{6}{ln}{p}_{i}≈22.5$, ${e}^{4.87}≈130$, and ${e}^{4.88}≈132$.

$(1)$ Among these six months, if two months are selected randomly, find the probability that one month has a carbon emission lower than the average emission of the six months, and the other month has a carbon emission higher than the average emission of the six months.

$(2)$ If a function model $p={p}_{0}{e}^{kt}$ is used to fit the two variables month $t$ and emission $p$, based on the data in the table, determine the regression equation of $p$ with respect to $t$.

Given: For the same set of data $(x_{1}$,$y_{1})$,$(x_{2}$,$y_{2})$,$\cdots $,$(x_{n}$,$y_{n})$, the least squares estimation formulas for the slope and intercept of the regression line $\hat{y}=\hat{b}x+\hat{a}$ are: $\hat{b}=\frac{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}({y}_{i}-\overline{y})}{\sum_{i=1}^{n}{{({x}_{i}-\overline{x})}^{2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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