
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  $ \alpha ,\beta $  be real numbers. Find the greatest value of the expression  $$  |\alpha x +\beta y| +|\alpha x-\beta y|  $$  in each of the following cases:**a)**  $ x,y\in \mathbb{R} $  and  $ |x|,|y|\le 1 $ **b)**  $ x,y\in \mathbb{C} $  and  $ |x|,|y|\le 1 $ 

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