
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 four-digit number $a b c d$ is called balanced if

$$
a+b=c+d
$$

Calculate the following quantities:

a) How many numbers $a b c d$ are there such that $a+b=c+d=8$?

b) How many numbers $a b c d$ are there such that $a+b=c+d=16$?

c) How many balanced numbers are there?

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