
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
In order to implement the requirements of "double reduction", enrich campus cultural life, and enhance class cohesion, the eighth grade of a certain school plans to hold the first "School $BA$ Class Basketball Game" with the theme of "Colorful June, Dreaming of Basketball" this year. The school plans to purchase several basketballs for the inter-class basketball game. After communicating with a sports goods dealer, the price of type $A$ basketballs is $40$ yuan more than the price of type $B$ basketballs. The number of type $A$ basketballs that can be purchased with $1200$ yuan is equal to the number of type $B$ basketballs that can be purchased with $600$ yuan.<br/>$(1)$ Find the unit prices of type $A$ and type $B$ basketballs, respectively.<br/>$(2)$ The sports goods store offers two discount options, and only one option can be chosen when purchasing;<br/>Option one: all items are sold at a $10\%$ discount;<br/>Option two: buy $3$ type $A$ basketballs and get $1$ type $B$ basketball for free (no free basketball if less than $3$ are purchased);<br/>If the school needs to buy $15$ type $A$ basketballs and $x\left(x\geqslant 5\right)$ type $B$ basketballs, which option is more cost-effective between the two purchasing options, and explain the reason.

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