
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
Currently, our city has set up three gas consumption tiers for gas charges for urban residents based on households and calculated annually as shown in the table below:

| Tier | Annual Gas Consumption | Sales Price | Note |
|------|------------------------|-------------|------|
| Tier 1 | Up to 400 $m^{3}$ (including 400) | $2.67$ yuan/$m^{3}$ | If the household has more than 4 people, the upper limits of gas consumption for Tier 1 and Tier 2 increase by 100 $m^{3}$ and 200 $m^{3}$ respectively for each additional person. |
| Tier 2 | 400 to 1200 $m^{3}$ (including 1200) | $3.15$ yuan/$m^{3}$ | |
| Tier 3 | Over 1200 $m^{3}$ | $3.63$ yuan/$m^{3}$ | |

$(1)$ For a household with 3 people and an annual gas consumption of 200 $m^{3}$, the gas fee for the year is ______ yuan;  
$(2)$ For a household with no more than 4 people and an annual gas consumption of $x m^{3}$ ($x > 1200$), the gas fee for the year is $y$ yuan. Find the functional expression of $y$ in terms of $x$;  
$(3)$ Household A has 3 people and Household B has 5 people. In a certain year, both households paid a gas fee of 3855 yuan. Find how many cubic meters of gas more Household B used than Household A in that year? (Round the result to 1 $m^{3}$)

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