
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 $f(x)=x$, $g(x)=x^2-x+3$. If there exist $x_1, x_2, \ldots, x_n \in [0, \frac{9}{2}]$ such that $f(x_{1})+f(x_{2})+\ldots +f(x_{n-1})+g(x_{n})=g(x_{1})+g(x_{2})+\ldots +g(x_{n-1})+f(x_{n})$, determine the maximum value of $n$.

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