
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 left and right foci $F_{1}$ and $F_{2}$, and a short axis length of $2$. The distance from the left vertex of ellipse $C$ to $F_{1}$ is $\sqrt{2}-1$.<br/>$(Ⅰ)$ Find the standard equation of ellipse $C$.<br/>$(Ⅱ)$ Let line $l$ intersect ellipse $C$ at points $A$ and $B$, with $M({0,\frac{1}{4}})$. If $\overrightarrow{MA}•\overrightarrow{MB}$ is a constant value, does the line $l$ pass through a fixed point? If it does, find the coordinates of the fixed point and the constant value; if it does not, explain why.

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