
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 hiker sees the peak of a mountain opposite them at an elevation angle of $\alpha = 23^{\circ} 20^{\prime}$ above the horizontal and sees the reflection of the same peak in the lake below at a depression angle of $\beta = 49^{\circ} 30^{\prime}$ below the horizontal. How high is the hiker above sea level if the mountain peak is $1827 \mathrm{~m}$ and the lake's surface is $608 \mathrm{~m}$ above sea level according to the map? How far is the hiker (straight-line distance) from the mountain peak?

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