
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
Define the function $f(x)$ on $\mathbb{R}$ as $f(x)=\begin{cases} \lg |x-2|, & x\ne 2 \\ 1, & x=2 \end{cases}$. If the equation $f^2(x)+bf(x)+c=0$ has exactly $5$ distinct real solutions $x_1, x_2, x_3, x_4, x_5$, calculate $f(x_1+x_2+x_3+x_4+x_5)$.

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