
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. If the standard weight of each bag is $400$ grams, and the differences between the weight of each bag and the standard weight are recorded in the table below:

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

$(1)$ Is the average weight of this batch of samples more or less than the standard weight? By how many grams?<br/>$(2)$ What is the total weight of the samples tested?

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