
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 shopping mall is holding a promotion where customers who spend a certain amount can participate in a ball-drawing activity to win rewards. The specific method is as follows: Two balls are drawn from a bag containing 2 red balls and 3 yellow balls (only differing in color). The reward amount is determined based on the number of red balls drawn, with the specific amounts set as shown in the table below:

| Number of Red Balls Drawn | 0   | 1   | 2   |
|---------------------------|-----|-----|-----|
| Reward (in yuan)         | 5   | 10  | 20  |

There are two ball-drawing schemes available:

Scheme 1: Draw one ball randomly, note the color, do not replace it, then draw another ball randomly from the remaining balls.

Scheme 2: Draw one ball randomly, note the color, replace it, then draw another ball randomly from all the balls.

$(1)$ Find the probability of drawing two red balls in Scheme 1.

$(2)$ Help customers analyze and choose which ball-drawing scheme is more advantageous from the perspective of average earnings.

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