
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 Cartesian coordinate system $xOy$, the distance from the moving point $M$ to the fixed point $F(1,0)$ is 1 greater than the distance to the $y$-axis.  
$(1)$ Find the equation of the trajectory of the moving point $M$;  
$(2)$ When $x\geqslant 0$, let the trajectory of the moving point $M$ be curve $C$. The line passing through $F$ intersects curve $C$ at points $P$ and $Q$. Lines $OP$, $OQ$, and the line $x=1$ intersect at points $A$ and $B$. Determine whether the circle with diameter $AB$ passes through the fixed point. If yes, find the coordinates of the fixed point; if not, explain the reason.

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