
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 $f(x)=A\sin(x+\varphi)$ ($A > 0$, $0 < \varphi < \pi$), and the maximum value of $f(x)$ for $x \in \mathbb{R}$ is $1$, its graph passes through the point $M\left( \frac{\pi}{3}, \frac{1}{2}\right)$.  
$(1)$ Find the analytical expression of $f(x)$.  
$(2)$ Given that $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$, and $f(\alpha)= \frac{3}{5}$, $f(\beta)= \frac{12}{13}$, find the value of $f(\alpha-\beta)$.

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