
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 city's industrial department plans to implement energy-saving and consumption-reducing technology upgrades for the small and medium-sized enterprises under its jurisdiction. The following is the result of a questionnaire survey on whether 400 enterprises support the technological transformation:<br/>

|          | Support | Not Support | Total |
|----------|---------|-------------|-------|
| Medium-sized Enterprises | 60 | 20 | 80 |
| Small Enterprises | 180 | 140 | 320 |
| Total | 240 | 160 | 400 |

$(1)$ Based on the small probability value $\alpha = 0.005$ of the independence test, can it be concluded that "supporting energy-saving and consumption-reducing technology upgrades" is related to "enterprise size"?<br/>
$(2)$ From the small and medium-sized enterprises that support the technological transformation above, 12 enterprises are randomly selected using stratified random sampling, and then 9 enterprises are randomly selected for rewards. Each medium-sized enterprise is rewarded with 60,000 yuan, and each small enterprise is rewarded with 20,000 yuan. Let $X$ be the total amount of rewards given (in 10,000 yuan). Find the distribution table and mean of $X$.<br/>
Given: ${\chi^2} = \frac{{n{{(ad-bc)}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$, $n=a+b+c+d$.<br/>

| $\alpha$ | 0.01 | 0.005 | 0.001 |
|----------|------|-------|-------|
| $\chi_{\alpha}$ | 6.635 | 7.879 | 10.828 |

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