
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
If ellipse $E_{1}$: $\dfrac{x^{2}}{a_{1}^{2}} + \dfrac{y^{2}}{b_{1}^{2}} = 1$ and ellipse $E_{2}$: $\dfrac{x^{2}}{a_{2}^{2}} + \dfrac{y^{2}}{b_{2}^{2}} = 1$ satisfy $\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} = m (m > 0)$, then these two ellipses are said to be similar, and $m$ is called the similarity ratio. If ellipse $M_{1}$ is similar to ellipse $M_{2}: x^{2} + 2y^{2} = 1$ and passes through the point $(1, \dfrac{\sqrt{2}}{2})$,
- $(I)$ Find the standard equation of ellipse $M_{1}$;
- $(II)$ A line $l$ with a non-zero slope passing through point $P(-2,0)$ intersects ellipse $M_{1}$ at two distinct points $A$ and $B$. $F$ is the right focus of ellipse $M_{1}$. Lines $AF$ and $BF$ intersect ellipse $M_{1}$ again at points $G$ and $H$, respectively. Let $\overrightarrow{AF} = \lambda_{1} \overrightarrow{FG}$ and $\overrightarrow{BF} = \lambda_{2} \overrightarrow{FH} (\lambda_{1}, \lambda_{2} \in \mathbb{R})$. Find the range of $\lambda_{1} + \lambda_{2}$.

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