
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
Three manufacturers, A, B, and C, have market shares of $25\%$, $35\%$, and $40%$ for their mobile phone chargers, respectively. The qualification rates of their chargers are $70\%$, $75\%$, and $80%$, respectively.
$(1)$ The local industrial and commercial quality inspection department randomly selects $3$ mobile phone chargers, with the number of chargers produced by Manufacturer A denoted as $X$. Find the probability distribution, expectation, and variance of $X$.
$(2)$ Now, one charger is randomly selected from the chargers produced by the three manufacturers, and it is found to be defective. Find the probability that it was produced by Manufacturer A.

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