
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 $f(\sqrt{2x+1})=x-\sqrt{2x+1}$.  
$(1)$ Find the analytical expression of the function $f(x)$;  
$(2)$ If the function $g(x)=\frac{2[f(x)-x+1]}{x}$, and the inequality $g(x)\leqslant ax$ holds for $x\in [\frac{1}{3}, 3]$, find the range of real number $a$.

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