
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 that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, and $P(1, \frac{\sqrt{2}}{2})$ is a point on the ellipse. It is also given that $\sqrt{2}|PF_{1}|$, $|F_{1}F_{2}|$, $\sqrt{2}|PF_{2}|$ form an arithmetic sequence.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Suppose a moving line $l$ passes through point $F_{2}$ and intersects the ellipse $C$ at points $A$ and $B$. Is there a fixed point $Q$ on the $x$-axis such that $\overrightarrow{QA} \cdot \overrightarrow{QB} = -\frac{7}{16}$ always holds? If it exists, find the coordinates of point $Q$; if not, explain why.

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