
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
Jiajia and Qiqi are playing a ball-touching game with 5 identical balls. Jiajia takes 3 balls and labels them with the numbers 2, 3, and 4 respectively. Qiqi takes 2 balls and also labels them with numbers. They put the balls into the same opaque bag and mix them evenly. Qiqi says, "The number I labeled is chosen from the numbers 3 and 4 (can be repeated)." After many ball-touching experiments, they found that the frequency of getting a ball with the number 3 is stable at 0.4.  
$(1)$ The mode of the numbers on these 5 balls is ______.  
$(2)$ After Qiqi mixes the balls in the bag, she draws a ball and says, "After drawing this ball, the median of the numbers on the remaining balls has not changed."  
① The number labeled on the ball Qiqi draws is ______.  
② Jiajia first draws a ball from the remaining balls, puts it back, mixes evenly, and then draws another ball. Calculate the probability that both balls Jiajia draws are even numbers, using a list or a tree diagram.

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