
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
Define a new operation \(x \, \star \, y\) for positive real numbers \(x\) and \(y\) where \(x \, \star \, y = x^y \cdot k\) for some fixed positive real number \(k\). Suppose the operation satisfies the equations \((x^y) \, \star \, y = x (y \, \star \, y)\) and \((x \, \star \, 1) \, \star \, x = x \, \star \, 1\) for all \(x, y > 0\). Given that \(1 \, \star \, 1 = k\), find the value of \(2 \, \star \, 3\).

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