
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
A certain factory has a fragile component $A$ on its machines. When this component is damaged during use, it needs to be sent to the repair department for maintenance. The factory stipulates that damaged component $A$ must be sent to the repair department before $8:30$ a.m. the next day, and the repair personnel must complete all maintenance work on damaged component $A$ on the same day. Each worker takes the same amount of time to independently repair component $A$. The repair department recorded the number of component $A$ repairs from the 1st to the 20th of a certain month, with the specific data shown in the table below:

| Date | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
|------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|
| Number of Component $A$ | 9 | 15 | 12 | 18 | 12 | 18 | 9 | 9 | 24 | 12 |

| Date | 11th | 12th | 13th | 14th | 15th | 16th | 17th | 18th | 19th | 20th |
|------|------|------|------|------|------|------|------|------|------|------|
| Number of Component $A$ | 12 | 24 | 15 | 15 | 15 | 12 | 15 | 15 | 15 | 24 |

One day is randomly selected from these 20 days, and the random variable $X$ represents the number of component $A$ repairs at the repair department on that day.

(I) Find the distribution table and mathematical expectation of $X$.

(II) If $a$, $b \in \mathbb{N}^{*}$ and $b-a=6$, find the maximum value of $P\left(a\leqslant X\leqslant b\right)$.

(III) Currently, there are two workers in the repair department engaged in maintenance work. To ensure that the mathematical expectation of the number of component $A$ repairs per worker does not exceed $4$ per day, how many additional repair workers are needed at least? (Only write down the conclusion).

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