
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 gym charges $20$ dollars per session. To reward customers, the gym has introduced the following promotional schemes: Scheme 1: Purchase a membership card for $40, and receive a 40% discount on each training session; Scheme 2: Do not purchase a membership card, and receive a 20% discount on each training session. Let $x$ represent the number of training sessions a customer attends. The cost under Scheme 1 is denoted as $y_{1}$ (in dollars), and the cost under Scheme 2 is denoted as $y_{2}$ (in dollars). <br/> $(1)$ Write down the relationship between $y_{1}$, $y_{2}$, and $x$ respectively; <br/> $(2)$ Xiao Li plans to attend the gym for training between $5$ and $20$ times. Which scheme should he choose to minimize the cost? Explain your reasoning.

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