
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 \( a, b \in \mathbb{Z}^+ \) and \( a < b \), if the pair of positive integers \( (a, b) \) satisfies:
\[ \mathrm{lcm}[a, b] + \mathrm{lcm}[a + 2, b + 2] = 2 \cdot \mathrm{lcm}[a + 1, b + 1], \]
then the pair \( (a, b) \) is called a "good array".

Find the largest constant \( c \) such that \( b > c \cdot a^3 \) holds for any good array \( (a, b) \).

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