
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 left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, respectively, and $P(1,\frac{\sqrt{2}}{2})$ is a point on the ellipse. It is also given that $\sqrt{2}|PF_1|$, $|F_1F_2|$, and $\sqrt{2}|PF_2|$ 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}
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