
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 points $F_1$ and $F_2$ are respectively the left and right foci of the ellipse $C_1$: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$), and point $F_2$ is also the focus of the parabola $C_2$: $y^{2}=8x$. Let $P$ be a moving point on the ellipse $C_1$, and the maximum area of $\triangle PF_1F_2$ is $2\sqrt{2}$.  
(Ⅰ) Find the equation of the ellipse $C_1$;  
(Ⅱ) Let $T$ be any point on the line $x=-3$, and draw a perpendicular line from $F_1$ to $TF_1$ intersecting the ellipse $C_1$ at points $M$ and $N$. Find the minimum value of $$\frac {|TF_{1}|}{|MN|}$$.

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