
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
During the "May Day" holiday, in order to meet the consumption needs of the people, a certain store plans to purchase a batch of household appliances for $160,000. The cost price and selling price of this batch of household appliances are as shown in the table below:

| Category | TV | Refrigerator | Washing Machine |
|----------|----|--------------|-----------------|
| Cost Price | $2000 | $1600 | $1000 |
| Selling Price | $2200 | $1800 | $1100 |

$(1)$ If all the funds are used to purchase a total of $100$ TVs and washing machines, how many of each can the store buy?

$(2)$ If within the current budget of $160,000, the store wants to purchase a total of $100$ units of the three types of household appliances in the table, with the number of TVs being the same as the number of refrigerators, and the number of washing machines not exceeding the number of TVs, calculate how many purchasing plans are possible. Which purchasing plan can maximize the profit for the store after selling these household appliances? What is the maximum profit? (Profit $=$ Selling Price $-$ Cost Price)

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