
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
We know that sound is generated by the vibration of objects and propagates in the form of waves in a certain medium (such as solids, liquids, gases). In physics, the energy of sound waves acting on a unit area perpendicular to their direction of propagation in unit time is called sound intensity $I$ ($W/cm^{2}$). However, in real life, the sound intensity level $D$ (in decibels, dB) is commonly used for measurement. To describe the functional relationship between sound intensity level $D$ (in dB) and sound intensity $I$ ($W/cm^{2}$), after multiple measurements, the following data is obtained:

| Group | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-------|---|---|---|---|---|---|---|
| Sound Intensity $I$ ($W/cm^{2}$) | $10^{-11}$ | $2\times 10^{-11}$ | $3\times 10^{-11}$ | $4\times 10^{-11}$ | $10^{-10}$ | ① | $9\times 10^{-7}$ |
| Sound Intensity Level $D$ (dB) | 10 | 13.01 | 14.77 | 16.02 | 20 | 40 | ② |

There are three function models to choose from: $D=kI+b$, $D=a\cdot I^{2}+c$, $D=m\lg I+n$.

(1) Based on the data from groups 1 to 5, select the function model that you think is more realistic, briefly explain the reason, and calculate the corresponding analytical expression based on the data from groups 1 and 5.

(2) Using the analytical expression obtained in (1) and the known data from the table, calculate the values for ① and ② in the table.

(3) Given that the noise level of fireworks is generally between (90,100) dB with sound intensity $I_{1}$, the noise level of firecrackers is generally between (100,110) dB with sound intensity $I_{2}$, and the noise level of airplane engine during takeoff is generally between (135,145) dB with sound intensity $I_{3}$. Determine the relationship between $I_{1}I_{3}$ and $I_{2}^{2}$, and explain the reason.

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