
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 $y=A\sin(\omega x+\varphi)$ ($A>0$, $\omega>0$, $|\varphi|<\frac{\pi}{2}$), its graph passes through point $P\left(\frac{\pi}{12}, 0\right)$, and the nearest highest point to $P$ on the graph has coordinates $\left(\frac{\pi}{3}, 5\right)$.  
(1) Find the analytical expression of the function;  
(2) Indicate the increasing interval of the function;  
(3) If the graph of this function is moved $\frac{\pi}{6}$ units to the left and then 2 units down to get the graph of $g(x)$, find the axis of symmetry of $g(x)$.

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