
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 $C: y^{2}=3x$, the line $l$ with a slope of $\frac{3}{2}$ intersects $C$ at points $E$ and $F$, and intersects the $x$-axis at point $H$. If $\overrightarrow{EH}=k\overrightarrow{HF}$ $(k \gt 1)$ and $|EF|=\frac{4\sqrt{13}}{3}$, calculate the value of $k$.

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