
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 factory produced an original calculator that performs two operations:

(a) the usual addition, denoted by \( + \) 

(b) an operation denoted by \( \circledast \).

We know that, for any natural number \( a \), the following hold:

\[ 
(i) \quad a \circledast a = a \quad \text{ and } \quad (ii) \quad a \circledast 0 = 2a 
\]

and, for any four natural numbers \( a, b, c, \) and \( d \), the following holds:

\[ 
(iii) \quad (a \circledast b) + (c \circledast d) = (a+c) \circledast(b+d) 
\]

What are the results of the operations \( (2+3) \circledast (0+3) \) and \( 1024 \circledast 48 \)?

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