
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's consider two positive real numbers $a$ and $b$, where an operation $a \, \blacktriangle \, b$ is defined such that $(ab) \, \blacktriangle \, b = a(b \, \blacktriangle \, b)$ and $(a \, \blacktriangle \, 1) \, \blacktriangle \, a = a \, \blacktriangle \, 1$ for all $a,b>0$. Additionally, it is given that $1 \, \blacktriangle \, 1 = 2$. Find the value of $23 \, \blacktriangle \, 45$.

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