
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
In the plane, the product of the slopes of the lines connecting point P with two fixed points $A_1(-a, 0)$ and $A_2(a, 0)$ (where $a > 0$) is a non-zero constant $m$. It is known that the eccentricity of the trajectory $C$ of point $P$ is $\frac{\sqrt{2}}{2}$.  
(1) Find the value of $m$;  
(2) Find the intersection points $A$ and $B$ of the ellipse $C$ with a line passing through the right focus of the ellipse $C$ and having a slope of $1$. If $O$ is the origin and $M$ is a point on the ellipse $C$ satisfying $\overrightarrow{OM} = \lambda \overrightarrow{OA} + \overrightarrow{OB}$, find the value of $\lambda$.

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