
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
Given that $x$ and $y$ are rational numbers, a new operation $*$ is defined such that $x*y=xy+1$.
- Find the value of $2*3$.
- Find the value of $(1*4)*\left(-\frac{1}{2}\right)$.
- Explore the relationship between $a*(b+c)$ and $a*b+a*c$, and express it using an equation.

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