
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 there is a particle in a coordinate plane starting from the origin and hopping along the x-axis. Each hop can be either 1 unit in the positive or negative direction. If after 5 hops the particle lands at the point (3,0) (allowing for the possibility that it passes over this point multiple times), then the total number of distinct methods of motion for the particle is _______. If after 20 hops the particle lands at the point (16,0) (also allowing for the possibility that it passes over this point multiple times), then the total number of distinct methods of motion for the particle is _______.

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