
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 food factory randomly selects $20$ bags of packaged food for testing whether the weight of each bag meets the standard. The differences between the weight of each bag and the standard weight are recorded in the table below:

| Difference from Standard Weight (in grams) | $-5$ | $-2$ | $0$ | $1$ | $3$ | $6$ |
|--------------------------------------------|------|------|----|----|----|----|
| Number of Bags                              | $1$  | $4$  | $3$| $4$| $5$| $3$|

$(1)$ If the standard weight for each bag is $450g$, what is the total weight of this batch of samples?<br/>$(2)$ Calculate the average weight of this batch of samples based on the condition in $(1)$, and determine whether it is heavier or lighter than the standard weight by how much?<br/>$(3)$ If the acceptable standard for this type of food is $450 \pm 5g$, what is the acceptance rate for this batch of samples?

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