
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 bag contains $4$ balls of the same size, numbered $1$, $2$, $3$, $4$. Now, a ball is drawn from the bag with replacement $2$ times.<br/>$(1)$ Find the probability of drawing the ball number $3$ both times;<br/>$(2)$ Let $X$ be the maximum value of the two draws. Find the distribution of $X$.

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