
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 particle begins its movement at the point $(3,5)$ and travels until it reaches the y-axis for the first time. At each position $(a,b)$, the particle can move randomly either to $(a-1,b)$ or $(a,b-1)$, with probability $\frac{1}{2}$ each. The probability that the particle hits the y-axis specifically at the point $(0,b)$ with $b > 0$, is expressed as $\frac{k}{2^m}$, where $k$ and $m$ are integers with $k$ not divisible by $2$. Calculate the value of $k + m$.

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