
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 four balls of the same shape and size, numbered $1$, $2$, $3$, and $4$.
$(1)$ Two balls are randomly drawn from the bag. Find the probability that the sum of the numbers on the drawn balls is not greater than $4$.
$(2)$ A ball is randomly drawn from the bag, its number recorded as $m$, and then returned to the bag. Another ball is drawn, and its number recorded as $n$. Find the probability that $n < m + 2$.

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