
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
For real numbers \( x_{1}, x_{2}, \cdots, x_{19} \), the maximum integer not exceeding the minimum value of
\[ f\left(x_{1}, x_{2}, \cdots, x_{19}\right)=\sum_{k_{1}=0}^{1} \sum_{k_{2}=0}^{1} \cdots \sum_{k_{19}=0}^{1}\left|k_{1} x_{1}+k_{2} x_{2}+\cdots+k_{19} x_{19}-1\right| \]
is ___ .

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