
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 two positive integers $a$ and $b$ satisfying $1 < a < b$, if the system of equations $\begin{cases} y=-2x+4033 \\ y=|x-1|+|x+a|+|x-b| \end{cases}$ has exactly one solution, then the maximum value of $a$ is \_\_\_\_\_\_.

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