
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 point starts from the origin on a line and makes $a$ steps to the right and $b$ steps to the left in some order, where $a > b$. The range of the point's walk is defined as the difference between the maximum and minimum coordinates of the point during its entire walk.

a) Find the maximum possible range of the walk.

b) Find the minimum possible range.

c) How many different sequences of the point's movements result in the maximum possible range of the walk?

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