
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
On a line, the points \( 0, \pm 1, \pm 2, \pm 3, \ldots \) are marked.

A particle, upon reaching point \( n \), moves to point \( n+1 \) with probability \( \frac{1}{2} \) and to point \( n-1 \) with probability \( \frac{1}{2} \) in one unit of time. Initially, the particle is at point 0. Find:

a) the probability \( x \) that the particle will ever be at point 1;

b) the probability \( y \) that the particle will ever be at point -1;

c) the probability \( z \) that the particle will ever return to point 0 (i.e., be at point 0 at a moment distinct from the initial one).

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