
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
The 2022 FIFA World Cup will be held in Qatar. A factory has designed a football souvenir and started production. The original plan was to produce 10,000 of these football souvenirs per day. However, due to various reasons, the actual daily production deviates from the planned quantity. The table below shows the production situation for a week (positive values indicate exceeding the plan, negative values indicate falling short of the plan, unit: pieces):

| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
|-----------------|--------|---------|-----------|----------|--------|----------|--------|
| Deviation from Plan | +41 | -34 | -52 | +127 | -72 | +36 | -29 |

$(1)$ According to the recorded data, how many more pieces were produced on the day with the highest production compared to the day with the lowest production this week?<br/>$(2)$ Did the actual total production for the week reach the planned quantity? Explain your reasoning.<br/>$(3)$ If the production cost of each football souvenir is $35$ yuan and they are sold for $40$ yuan each, what is the total profit for the factory this week?

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