
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 brand of hand sanitizer has a standard weight of $500$ grams per bottle. Twenty bottles of hand sanitizer were sampled from the production line to test whether each bottle's weight meets the standard. The differences between the weight of each bottle and the standard weight are recorded in the table below:

| Difference from Standard Weight (grams) | $-4$ | $-2$ | $0$ | $1$ | $3$ | $5$ |
|----------------------------------------|------|------|----|----|----|----|
| Number of Bottles                       | $1$  | $4$  | $3$| $4$| $5$| $3$|

$(1)$ If it is defined that a difference within $3$ grams (inclusive) of the standard weight is considered acceptable, how many bottles out of the $20$ sampled are considered acceptable?

$(2)$ Is the total weight of this batch of samples more or less than the standard weight? By how many grams?

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