
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 an ellipse $C: \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ $(a>b>0)$ with eccentricity $e=\frac{\sqrt{2}}{2}$, a circle centered at the origin with the radius equal to the semi-minor axis of the ellipse touches the line $x-y+\sqrt{2}=0$.
(Ⅰ) Find the equation of the ellipse $C$.
(Ⅱ) If a line passing through point $M(2,0)$ intersects the ellipse $C$ at two points $A$ and $B$, and there exists a point $P$ on the ellipse such that $\overrightarrow{OA}+\overrightarrow{OB}=t\overrightarrow{OP}$ (where $O$ is the origin), find the maximum integer value of $t$.

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