
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 certain company occasionally purchases two types of pre-made foods produced by a certain factory: mixed sauce noodles and beef noodles.  
$(1)$ The company spent $3000$ yuan to purchase a total of $170$ portions of mixed sauce noodles and beef noodles. At that time, the prices of mixed sauce noodles and beef noodles were $15$ yuan and $20$ yuan, respectively. Find out how many portions of each type of food were purchased.  
$(2)$ Due to adjustments in the number of company employees and food prices, the company now spends $1260$ yuan and $1200$ yuan respectively to purchase mixed sauce noodles and beef noodles. It is known that the number of portions of mixed sauce noodles purchased is $50\%$ more than the number of portions of beef noodles, and each portion of mixed sauce noodles costs $6$ yuan less than each portion of beef noodles. Find out how many portions of beef noodles were purchased.

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