
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ passing through point $A(-2,0)$, with its right focus at $F(1,0)$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let $P$ be a moving point on the ellipse $C$ (not on the $x$-axis), $M$ be the midpoint of $AP$, a line parallel to $AP$ passing through the origin $O$ intersects the line $x=t$ $(t>1)$ at point $Q$. Determine if $t$ can be a constant value such that $OM \perp FQ$. If it is a constant value, find the value of $t$; if not, explain the reason.

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