
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
Let $f(x)=e^{2x}+a$, $g(x)=e^{x}+x$. If there exist $x_{1}$, $x_{2}$, $\ldots$, $x_{2023}\in \left[-1,1\right]$, such that $\sum_{i=1}^{2022}f(x_{i})+g(x_{2023})=\sum_{i=1}^{2022}g(x_{i})+f(x_{2023})$ holds, then the maximum value of the real number $a$ is ____.

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