
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
To promote water conservation, a certain place implements a tiered water price system by month, with prices as shown in the table below $(m^{3}$ represents cubic meters):

| Tier       | Monthly Water Usage $(m^{3})$ | Price (元/$m^{3}$) |
|------------|-------------------------------|---------------------|
| Tier 1     | Up to $22$                    | $3$                 |
| Tier 2     | Over $22$ but up to $30$      | $5$                 |
| Tier 3     | Over $30$                     | $7$                 |

$(1)$ If a household, A, uses a total of $25m^{3}$ of water in April, the water bill should be ______ yuan;  
$(2)$ Let the monthly water usage be $x m^{3}$, and the water bill for the month be $y$ yuan. When $x \gt 30$, $y=$ ______ (expressed in a formula containing $x$); Use the above formula to calculate: If household B paid a total of $120$ yuan for water in May, the water usage was ______ $m^{3}$;  
$(3)$ If household C uses a total of $50m^{3}$ of water in May and June (with less water used in May than in June), and the total water bill for these two months is $174$ yuan, then what are the water usages for household C in May and June respectively?

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