
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 $1$, $2$, $3$, $4$ respectively.
$(1)$ Randomly draw two balls from the bag, and calculate the probability that the sum of the numbers on the drawn balls is even.
$(2)$ First, randomly draw a ball from the bag, and record its number as $m$. Then, return the ball to the bag and draw another ball, recording its number as $n$. Calculate the probability that $n < m+1$.

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