
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 certain city implements a tiered electricity price for residents' daily electricity usage, with specific charging standards as shown in the table below:

| Tier | Monthly Electricity Usage | Price (元/$) |
|------|---------------------------|--------------|
| Tier 1 | Up to 240 degrees | $a$ |
| Tier 2 | Over 240 degrees but not exceeding 400 degrees | $0.65$ |
| Tier 3 | Exceeding 400 degrees | $a+0.3$ |

It is known that in October, Mr. Li's household in the city used 200 degrees of electricity and paid $120. In September, Mr. Li's household paid $157 for electricity.

$(1)$ The value of $a$ in the table is ______.

$(2)$ Find the electricity usage of Mr. Li's household in September.

$(3)$ If the average electricity price for Mr. Li's household in August was $0.7$ yuan per degree, find the electricity usage of Mr. Li's household in August.

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