
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
To understand the delivery timeliness and service satisfaction of customers for two express delivery companies, $A$ and $B$, a random survey of customer evaluations for these two companies was conducted in a certain region. It is known that there are 120 and 80 survey questionnaires for companies $A$ and $B$ respectively. The data is summarized as follows:

| Express Company | $A$ Express Company | $B$ Express Company |
|-----------------|----------------------|----------------------|
| Item/Score Range | Delivery Timeliness | Service Satisfaction | Delivery Timeliness | Service Satisfaction |
| $85\leqslant x\leqslant 95$ | 29 | 24 | 16 | 12 |
| $75\leqslant x \lt 85$ | 47 | 56 | 40 | 48 |
| $65\leqslant x \lt 75$ | 44 | 40 | 24 | 20 |

Assume that customer evaluations for companies $A$ and $B$ are independent. Using frequency estimation to calculate probabilities:

$(1)$ From customers who choose company $A$ in this region, if one person is randomly selected, estimate the probability that this customer's evaluation of the delivery timeliness of company $A$ is at least 75 points.

$(2)$ From the sample survey questionnaires of companies $A$ and $B$ in this region, if one questionnaire is randomly selected from each, let $X$ be the number of questionnaires with service satisfaction ratings of at least 75 points. Find the distribution table and mathematical expectation of $X$.

$(3)$ Let the evaluation score $x\geqslant 85$ be classified as "Excellent," $75\leqslant x \lt 85$ as "Good," and $65\leqslant x \lt 75$ as "Fair." Given that Xiao Wang values delivery timeliness ratings, based on the sample evaluation scores of delivery timeliness for companies $A$ and $B$ in this region, which company do you think Xiao Wang should choose? Provide reasons.

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