
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 proton is known to move on the number line from the origin under the influence of random external forces, moving one unit in the positive or negative direction of the number line with equal probability every second. If it moves $n$ times, then when $n=4$, the probability that the proton is at the origin is ______, and when $n=$______, the probability that the proton is at the point corresponding to $6$ is maximized.

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