
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 the function $g(x)=a\sin x+b\cos x+c$.
(1) Find the range of $g(x)$ when $b=0$.
(2) When $a=1$, $c=0$, the graph of function $g(x)$ is symmetric about $x=\frac{5π}{3}$. Find the axis of symmetry of the function $y=b\sin x+a\cos x$.
(3) If there is a lowest point $(\frac{11π}{6},1)$ on the graph of $g(x)$, and if the graph is transformed such that each point's vertical coordinate remains unchanged, the horizontal coordinate is shortened to $\frac{3}{π}$ times its original length, and then shifted left by 1 unit to obtain the graph of $y=f(x)$, and if all positive roots of $f(x)=3$ are $x_{1}$, $x_{2}$, $x_{3}$, ..., $x_{n}$, ... in ascending order, and $x_{n}-x_{n-1}=3(n\geqslant 2)$, find the analytical expression of $f(x)$.

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