
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 business hall launched two package plans on New Year's Day, with specific billing methods as shown in the table below:

| Package | Monthly Basic Fee | Outgoing Call Limit | Outgoing Call Overtime Fee | Incoming Call |
|---------|-------------------|---------------------|----------------------------|--------------|
| Plan 1  | $58$ yuan          | $150$ minutes       | $0.25$ yuan per minute     | Free         |
| Plan 2  | $88$ yuan          | $350$ minutes       | $a$ yuan per minute        | Free         |

$(1)$ If the outgoing call time is $260$ minutes, the cost of choosing Plan 1 is ______ yuan, and the cost of choosing Plan 2 is ______ yuan.  
$(2)$ If $a=0.3$ in the table, please discuss and explain whether there exists an outgoing call time $t$ that makes the billing of the two plans equal.  
$(3)$ If the billing of the two plans is equal when the outgoing call time is $450$ minutes, then $a=______$. In this case, when the outgoing call time $t$ satisfies the condition ______, choosing Plan 1 saves more money.

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