
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 $f(x)=A\sin (\omega x+\varphi)$ ($A > 0$, $\omega > 0$, $0 < \varphi < \pi$), $x\in\mathbb{R}$, its maximum value is $1$, and its minimum positive period is $2\pi$, and its graph passes through point $M(0,1)$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ Let $A$, $B$, $C$ be the three interior angles of $\triangle ABC$, and $f(A)= \frac {3}{5}$, $f(B)= \frac {5}{13}$, $a=13$, find the area of $\triangle ABC$.

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