
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
Determine the distance between the two stars in the Big Dipper that appear to be the farthest apart - astronomically denoted as $\alpha \mathrm{U}$ Ma and $\eta \mathrm{U}$ Ma - given their distances from the Sun are 59, respectively 200 light years, and their celestial coordinates (declination and right ascension) are:

$$
\begin{array}{lll}
\alpha \text { U Ma case: } & \delta=+62^{\circ} 9^{\prime}, & a=10^{\mathrm{h}} 59^{\mathrm{m}} \\
\eta \text { U Ma case: } & \delta=+49^{\circ} 41^{\prime}, & a=13^{\mathrm{h}} 45^{\mathrm{m}}
\end{array}
$$

Also calculate the distance between the two stars in the Little Dipper that appear to be the farthest apart. The data:
$\beta$ U Mi: dist.: 96 light years, $\delta=74^{\circ} 28^{\prime}, \quad a=14^{\mathrm{h}} 51^{\mathrm{m}}$.

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