
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 the function $f(x) =
\begin{cases} 
\log|x-2| & \text{if } x \neq 2 \\
1 & \text{if } x=2
\end{cases}
$, find $f(x_1 + x_2 + x_3 + x_4 + x_5)$ if the equation $f(x)^2 + bf(x) + c = 0$ has exactly five distinct real roots $x_1, x_2, x_3, x_4, x_5$.

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