
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
Can the area of a triangle be determined by the lengths of its three sides?<br/>The Heron's formula tells you the calculation method: $S=\sqrt{p(p-a)(p-b)(p-c)}$, where $S$ represents the area of the triangle, $a$, $b$, $c$ represent the lengths of the three sides, and $p$ represents half of the perimeter, i.e., $p=\frac{a+b+c}{2}$.<br/>The "Method of Triangulation for Calculating Areas" proposed by the Chinese mathematician Qin Jiushao in the Song Dynasty is basically the same as this formula, so this formula is also called the "Heron-Qin Jiushao formula".<br/>Please use the formula to answer the following questions.<br/>$(1)$ In $\triangle ABC$, given $AB=5$, $BC=6$, $CA=7$, find the area of $\triangle ABC$;<br/>$(2)$ Calculate the height on side $BC$ of $\triangle ABC$ from question $(1)$.

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