
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
Suppose a particle on the x-axis starts at the origin. Assume that at every unit of time, it is subjected to a random force causing its position to change, moving either one unit to the right with probability \( p \) or one unit to the left with probability \( q = 1 - p \). The particle can only be at integer points on the x-axis. Find the probability that the particle is at integer point \( k \) at time \( t = n \).

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