
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 noodle shop offers classic specialty noodles to customers, who can either dine in at the shop (referred to as "dine-in" noodles) or purchase packaged fresh noodles with condiments (referred to as "fresh" noodles). It is known that the total price of 3 portions of "dine-in" noodles and 2 portions of "fresh" noodles is 31 yuan, and the total price of 4 portions of "dine-in" noodles and 1 portion of "fresh" noodles is 33 yuan.  
$(1)$ Find the price of each portion of "dine-in" noodles and "fresh" noodles, respectively.  
$(2)$ In April, the shop sold 2500 portions of "dine-in" noodles and 1500 portions of "fresh" noodles. To thank the customers, starting from May 1st, the price of each portion of "dine-in" noodles remains unchanged, while the price of each portion of "fresh" noodles decreases by $\frac{3}{4}a\%$. After analyzing the sales and revenue in May, it was found that the sales volume of "dine-in" noodles remained the same as in April, the sales volume of "fresh" noodles increased by $\frac{5}{2}a\%$ based on April, and the total sales of these two types of noodles increased by $\frac{1}{2}a\%$ based on April. Find the value of $a$.

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