
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
Let \( a > b > 1 \) be relatively prime positive integers. Define the weight of an integer \( c \), denoted by \( w(c) \), to be the minimal possible value of \( |x| + |y| \) taken over all pairs of integers \( x \) and \( y \) such that
\[ 
a x + b y = c .
\]

An integer \( c \) is called a local champion if \( w(c) \geq w(c \pm a) \) and \( w(c) \geq w(c \pm b) \). Find all local champions and determine their number.

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