
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 for any \( x_{1}, x_{2}, \cdots, x_{2020} \in [0, 4] \),

the equation
$$
\left|x - x_{1}\right| + \left|x - x_{2}\right| + \cdots + \left|x - x_{2020}\right| = 2020a
$$

has at least one root in the interval \([0, 4]\), find the value of \( a \).

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