
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
On December 9th, the Dongfang Selection Jilin Special officially started broadcasting, with hundreds of Jilin specialties quickly selling out in the Douyin live broadcast room, including Korean-style spicy cabbage, Haoyue beef, frozen pears, rice, and more. After 5 hours of broadcasting, the number of orders exceeded 410,000. The first link in the live broadcast room is for Haoyue beef, priced at $100 per order; the second link is for spicy cabbage, priced at $30 per order; and the third link is for a combination of beef and spicy cabbage, priced at $118 per order. 

$(1)$ A certain unit canteen plans to purchase 50 orders of beef and $a$ orders of spicy cabbage $\left(a \gt 50\right)$. If purchased through the first and second links, due to the large order quantity, spicy cabbage and beef can be discounted by 10%. The amount to be paid is ______ yuan. If purchased through the second and third links, as the combination package is already discounted, spicy cabbage and beef will not be discounted. The amount to be paid is represented by an algebraic expression containing $a$.

$(2)$ Under the conditions of $(1)$, if purchasing 50 orders of beef and 70 orders of spicy cabbage, explain through calculations which method of purchase is the most cost-effective.

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