
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 set $A$ consist of all ordered pairs of real numbers. Define an operation on $A$, denoted as $⊙$, such that for any two elements $α=(a,b)$, $β=(c,d)$ in $A$, the operation is defined as: $α⊙β=(ad+bc,bd-ac)$.

(I) Calculate: $(2,3)⊙(-1,4)$;

(II) Express the commutative law of operation $⊙$ using mathematical symbolic language and provide a proof;

(III) If $I∈A$, and $∀α∈A$, it follows that $I⊙α=α$, find the element $I$.

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