
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 $a_{1}$ in the sequence $\{a_{n}\}$, and the function $f(x)=\frac{1}{3}{x^3}-{a_{n+1}}sinx+({{a_n}+2})x+1$ has a unique zero for its derivative, and the function $g\left(x\right)=8x+\sin \left(\pi x\right)-\cos \left(\pi x\right)$ with $g(a_{1})+g(a_{2})+\ldots +g(a_{9})=18$. Find $a_{5}$.

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