
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 $M(1,2)$ is a point on the parabola $C:y^{2}=2px$.  
$(1)$ Find the directrix equation of the parabola $C$.  
$(2)$ A line $l$ passing through point $T(0,1)$ intersects the parabola $C$ at points $A$ and $B$, and the slopes of lines $MA$ and $MB$ are complementary. Find the value of $|TA|\cdot |TB|$.

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