
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
For the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1$, with $F_{1}$ and $F_{2}$ as its left and right foci, and $P(m,n)$ as a point on the ellipse different from the vertices. Let $\angle PF_{1}F_{2}=\alpha$ and $\angle PF_{2}F_{1}=\beta$, the correct conclusions are \_\_\_\_\_\_  
(1) If $\triangle PF_{1}F_{2}$ is an acute triangle, then $\sin \alpha < \cos \beta$.  
(2) The eccentricity of the ellipse $e= \dfrac {\sin (\alpha+\beta)}{\sin \alpha+\sin \beta}$.  
(3) If $\triangle PF_{1}F_{2}$ is an acute triangle, then the ratio of the distances from its circumcenter to the three sides is $\sin \alpha$ : $\sin \beta$ : $\sin (\alpha+\beta)$.  
(4) There exists a fixed circle that is tangent to the circle with $P$ as its center and $PF_{2}$ as its radius.  
(5) $\dfrac {1}{m^{2}}+ \dfrac {1}{n^{2}}\geqslant ( \dfrac {1}{a}+ \dfrac {1}{b})^{2}$.

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