
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 bag contains $5$ balls of the same size and shape, numbered as follows: $(1,1,2,2,3)$.

$(1)$ Calculate the probability of drawing a ball numbered $3$ when $3$ balls are randomly drawn from the bag without replacement.

$(2)$ Now, $3$ balls are randomly drawn from the bag with replacement, and this process is repeated $10$ times. Let $X$ denote the number of times a ball numbered $3$ is drawn. Calculate the mathematical expectation and variance of $X$.

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