
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 ellipse $C_1$: $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ ($a>b>0$) with its right vertex at $A(1,0)$, and a chord passing through the foci of $C_1$ and perpendicular to the major axis has a length of 1.  
(Ⅰ) Find the equation of the ellipse $C_1$;  
(Ⅱ) Let point $P$ be on the parabola $C_2$: $y=x^{2}+h$ ($h\in \mathbb{R}$), and the tangent line to $C_2$ at point $P$ intersects $C_1$ at points $M$ and $N$. When the x-coordinate of the midpoint of segment $AP$ equals the x-coordinate of the midpoint of segment $MN$, find the minimum value of $h$.

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