
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
Alice and Bob live 12 miles apart. One day, Alice looks due north from her house and sees an airplane. At the same time, Bob looks due north-east from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $45^\circ$ from Bob's position. Determine the altitude of the airplane, in miles.

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