
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 one of its foci at $F(1,0)$ and an eccentricity of $\frac{\sqrt{2}}{2}$.
(1) Find the equation of ellipse $C$.
(2) Line $y=x+m$ intersects $C$ at points $A$ and $B$. If $\overrightarrow{OA}\cdot \overrightarrow{OB}=-1$, where $O$ is the origin, find the value of the real number $m$.

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