
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
With the development of technology, various apps are emerging on mobile phones. Among them, Douyin is a very popular self-media software that helps users express themselves and record a beautiful life on a video platform. While most people use it for entertainment, some people with business acumen use Douyin for live streaming sales, making a lot of money. The prices of goods on Douyin change with the popularity of the videos. It is calculated that the price of a certain clothing item approximately satisfies the formula: $J={J_b}+({{J_0}-{J_b}}){({\frac{1}{2}})^{\frac{t}{h}}}$, where $J_{0}$ (unit: yuan) represents the initial price of the clothing item, $J$ (unit: yuan) represents the price after a certain time $t$ (unit: days), $J_{b}$ (unit: yuan) represents the fluctuation price, and $h$ (unit: days) represents the fluctuation period. A merchant sells this clothing item through Douyin, starting at a price of $120$ yuan per item, with a fluctuation price of $20$ yuan per item. It takes $10$ days for the price of the clothing item to drop to $70$ yuan per item.<br/>$(1)$ Find the value of $h$;<br/>$(2)$ Find the number of days needed for the price of the clothing item to drop to $60$ yuan per item. (Round the result to the nearest integer)<br/>Reference data: $\lg 2\approx 0.3010$

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