
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
If the function $f(x)$ defined on $\mathbb{R}$ is given by
$$
f(x)=
\begin{cases}
\lg |x-2|, & \text{if } x\neq 2, \\
1, & \text{if } x=2,
\end{cases}
$$
find the value of $f(x_1+x_2+x_3+x_4+x_5)$, where $x_1, x_2, x_3, x_4, x_5$ are the roots of the equation $f^2(x) + bf(x) + c = 0$ and the equation has exactly $5$ distinct real solutions.

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