
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 "$5A$" scenic area decided to launch discounted packages during the Labor Day period of "$5.1$". They offered pre-sale tickets for "Parent-child Tour for Two" and "Family Tour for Three". The price of the pre-sale "Family Tour for Three" package is twice the price of the "Parent-child Tour for Two" package.<br/>$(1)$ If the pre-sale amount for the "Parent-child Tour for Two" package is $21,000$ yuan, and the pre-sale amount for the "Family Tour for Three" package is $10,500$ yuan, with the sales volume of the "Parent-child Tour for Two" being $450$ more than the "Family Tour for Three", find the price of the "Parent-child Tour for Two" package.<br/>$(2)$ On the day of sale, the plan was to introduce $1,600$ tickets for the "Parent-child Tour for Two" and $400$ tickets for the "Family Tour for Three". Due to the hot pre-sale, the scenic area decided to increase the price of the "Parent-child Tour for Two" package by $\frac{3}{4}a$ yuan based on the price in (1), while the price of the "Family Tour for Three" package increased by $a$ yuan from the price in (1). As a result, the sales volume of the "Parent-child Tour for Two" package was $32a$ less than planned, while the sales volume of the "Family Tour for Three" package remained consistent with the plan. The final actual sales amount was the same as the planned sales amount. Find the value of $a$.

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