
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
Calculate the value of $K$ when $x, y,$ and $z$ are replaced by the first, second, and third numbers of the sets I-V, respectively. Then factor $K$ and explain the observations from the calculations.

$$
\begin{aligned}
K & =(x+2 y+3 z)(2 x-y-z)(y+2 z+3 x)+ \\
& +(y+2 z+3 x)(2 y-z-x)(z+2 x+3 y)+ \\
& +(z+2 x+3 y)(2 z-x-y)(x+2 y+3 z)
\end{aligned}
$$

I. $1,4,9 ; \quad$ II. $11,14,19 ; \quad$ III. $4,9,1$; IV. $1,8,11 ; \quad$ V. $5,8,-2$.

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