
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 a non-zero sequence \(\{a_n\}\) that satisfies \(a_1 = 1\), \(a_2 = 3\), and \(a_n (a_{n+2} - 1) = a_{n+1} (a_{n+1} - 1)\) for \(n \in \mathbf{N}^*\), find the value of \(\mathrm{C}_{2023}^{0} a_1 + \mathrm{C}_{2023}^{1} a_2 + \mathrm{C}_{2023}^{2} a_3 + \cdots + \mathrm{C}_{2023}^{2023} a_{2024}\) expressed as an integer exponentiation.

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