
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
A factory's waste gas is discharged after filtration. The relationship between the pollutant content $P$ (unit: $mg/L$) of the waste gas and time $t$ (unit: $h$) during the filtration process is given by $P={P}_{0}{e}^{-kt}$, where $P_{0}$ and $k$ are positive constants. It is known that $10\%$ of the pollutant was eliminated in the first $5h$.

$(1)$ What percentage of the pollutant remains after $10h$?

$(2)$ How much time is needed at least to reduce the pollutant by more than one-third? (Round the result to the nearest hour)

(Given data: $\ln 2=0.69, \ln 3=1.1, \ln 5=1.61$)

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