
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 that the right focus of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$) is $F_{2}(1,0)$, and the point $H(2,\frac{2\sqrt{10}}{3})$ is on the ellipse.

(1) Find the equation of the ellipse;
(2) Point $M$ is on the circle $x^{2}+y^{2}=b^{2}$, and $M$ is in the first quadrant. A tangent line is drawn from $M$ to the circle $x^{2}+y^{2}=b^{2}$, which intersects the ellipse at points $P$ and $Q$. Is the perimeter of triangle $PF_{2}Q$ a constant value? If so, find this constant value; if not, explain why.

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