
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 movie "Water Gate Bridge of Changjin Lake" released during last year's Spring Festival vividly interprets the great spirit of the Chinese people's resistance against the United States and aid to Korea. In order to conduct patriotic education for students, a certain school has organized a "Love for China" classic recitation activity and set up first, second, and third prizes. They need to purchase 50 prizes in total. The number of second prize items is 2 less than 3 times the number of first prize items. The unit prices of the various prizes are as shown in the table below:

|            | First Prize | Second Prize | Third Prize |
|------------|-------------|--------------|-------------|
| Unit Price (¥/item) | $20$ | $14$ | $8$ |
| Quantity (items) | $x$ | ______ | ______ |

$(1)$ Express the number of second prize items and third prize items in terms of $x$.  
$(2)$ Find the total cost of purchasing these 50 prizes in terms of $x$ in its simplest form.  
$(3)$ If 22 second prize items are purchased, how much money did the school spend in total?

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