
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
Determine for which values of \(a\), the system \(x + y - 144 = 0\) and \(xy - 5184 - 0.1 a^2 = 0\) has one solution. 

Find the smallest number among natural numbers greater than 2022 that divides \(2021!! + 2022!!\) exactly. (The symbol \( n!! \) denotes the product of all natural numbers not exceeding \(n\) and having the same parity: \( n!! = n \cdot (n-2) \cdot (n-4) \ldots) \).

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