
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 small factory produces handheld fans in blue and pink colors, with a daily production of 200 units. Among them, 120 units are blue handheld fans and 80 units are pink handheld fans.
$(1)$ If 2 handheld fans are randomly sampled from the production of handheld fans on a certain day, with $X$ representing the number of blue handheld fans sampled, both with replacement and without replacement, find the distribution and mathematical expectation of $X$.
$(2)$ If 10 handheld fans are randomly sampled from the production on a certain day, with $Y$ representing the number of blue handheld fans in the sample, estimate the proportion of blue handheld fans in the total population using the proportion of blue handheld fans in the sample. Find the probability that the error does not exceed 0.1, and determine which sampling method provides a more reliable estimate under the same error limit.
Reference data: Probability values for the binomial distribution and hypergeometric distribution corresponding to the random variable $Y$ (accurate to $0.00001$).

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