
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 random walk of a point starts from the origin on a line and makes \(a\) steps to the right, \(b\) steps to the left in some order, with \(a > b\). The range of the walk is defined as the difference between the highest and lowest coordinates of the point during the walk.

a) Find the maximum possible range of the walk.

b) Find the minimum possible range of the walk.

c) How many different sequences of movements are there in which the range of the walk is the maximum possible?

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