
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 15 miles apart. One day, Alice looks due north from her house and sees an airplane. At the same time, Bob looks due west from his house and also sees the same airplane. The angle of elevation of the airplane from Alice's position is $45^\circ$ and from Bob's position is $45^\circ$. Determine the closest to the airplane's altitude, in miles.

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