
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 the parabola $E$: $x^{2}=4y$ with focus $F$, and point $P(a,0)$ on the $x$-axis.
$(1)$ When $a\neq 0$, a line $l$ passes through point $P$ and is tangent to $E$. Find the equation of the tangent line $l$;
$(2)$ There exist two lines $l_{1}$ and $l_{2}$ passing through point $P$ with complementary inclination angles. If $l_{1}$ and $l_{2}$ intersect $E$ at points $A$, $B$, and $C$, $D$ respectively, and the areas of $\triangle FAB$ and $\triangle FCD$ are equal, find the range of the real number $a$.

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