
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
According to the table, answer the following questions:

| $x$   | $13$   | $13.1$ | $13.2$ | $13.3$ | $13.4$ | $13.5$ | $13.6$ | $13.7$ | $13.8$ | $13.9$ | $14$  |
|-------|--------|--------|--------|--------|--------|--------|--------|--------|--------|--------|-------|
| $x^{2}$ | $169$  | $171.61$ | $174.24$ | $176.89$ | $179.56$ | $182.25$ | $184.96$ | $187.69$ | $190.44$ | $193.21$ | $196$ |

$(1)$ The square root of $190.44$ is ______.

$(2)$ $\sqrt{176.9}\approx $______, $\sqrt{18769}=\_\_\_\_\_\_$.

$(3)$ If $13.5 \lt \sqrt{n} \lt 13.6$, find the value of the integer $n$ that satisfies the condition.

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