
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
"The purpose of playing happily and safely" is the working principle of various scenic spots. A certain scenic spot opens its doors to welcome visitors at $8:00$ in the morning and prohibits visitors from entering at $5:00$ in the afternoon. According to the staff statistics, at $9:00$ in the morning, the scenic spot had already accumulated $950$ visitors, and visitors who finished playing started to leave from that time. The data on the changes in the cumulative number of visitors entering the scenic spot $y$ (unit: people) and the cumulative number of visitors leaving the scenic spot $z$ (unit: people) with the statistical time $x$ (unit: hours) are shown in the table below:

| Statistical Time $x$/h | $1$ | $2$ | $3$ | $4$ | $\ldots$ |
|------------------------|-----|-----|-----|-----|----------|
| Cumulative Number of Visitors Entering the Scenic Spot $y$/people | $950$ | $1800$ | $2650$ | $3500$ | $\ldots$ |
| Cumulative Number of Visitors Leaving the Scenic Spot $z$/people | $0$ | $200$ | $400$ | $600$ | $\ldots$ |

It was found that the quantitative relationship between $y$ and $x$, and $z$ and $x can be described using functions we have learned.

$(1)$ Write down the analytical expression of the function of $y$ with respect to $x$ and the function of $z$ with respect to $x$ directly (no need to specify the range of the independent variable).

$(2)$ At what time is the number of visitors in the scenic spot expected to be the highest?

$(3)$ When the number of visitors in the scenic spot reaches $2600$, it will trigger a yellow warning for peak flow of people. At what time will the yellow warning for peak flow of people be triggered? Provide the answer directly.

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