
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
In an opaque bag, there are $5$ yellow balls, $13$ black balls, and $22$ red balls, all of which are identical except for their colors.  
$(1)$ Xiao Ming and Xiao Hong are playing a ball-touching game, where each player touches a ball and then returns it to the bag. If a player touches a black ball, Xiao Ming wins; if a player touches a yellow ball, Xiao Hong wins. Is this game fair for both players? Please explain your reasoning.  
$(2)$ Now the referee wants to take out some black balls from the bag and replace them with the same number of yellow balls to make the game fair for both players. How many black balls should be taken out?

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