
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 \( f \) be a real-valued function with the rule \( f(x) = x^{3} + 3x^{2} + 6x + 14 \) defined for all real values of \( x \). It is given that \( a \) and \( b \) are two real numbers such that \( f(a) = 1 \) and \( f(b) = 19 \). Find the value of \( (a + b)^{2} \).

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