
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 furniture store plans to purchase a certain type of dining table and dining chairs for sale, with the following information:

|            | Original Purchase Price (in yuan per piece) | Retail Price (in yuan per piece) | Set Price (in yuan per set) |
|------------|--------------------------------------------|----------------------------------|-----------------------------|
| Dining Table | $a$ | $270$ | $500$ |
| Dining Chair | $a-110$ | $70$ | |

It is known that the number of dining tables purchased for $600$ yuan is the same as the number of dining chairs purchased for $160$ yuan.

(1) Find the value of $a$ in the table.

(2) If the store purchases the dining chairs in a quantity that is five times the quantity of dining tables plus 20 more chairs, and the total quantity of dining tables and chairs does not exceed 200 pieces. The store plans to sell half of the dining tables as sets (one table and four chairs as a set) and the remaining tables and chairs at retail. How should the store purchase the items to maximize profit? What is the maximum profit?

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