
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

If the beam \( EC' \) reaches the eyepiece after reflections in the mirrors, meaning the direction of the beam after two deviations aligns with the direction \( DA \), then the total deviation of the beam \( EC' \) is equal to the angle between the beams \( EC' \) and \( DA \):

$$
\psi_{\text{total}} = 2\phi = \alpha
$$

From which it follows that

$$
\phi = 0.5 \alpha = 15^{\circ}
$$

Thus, we have shown that if the beam \( EC' \) after deviations coincides with the beam \( DA \), then the angle between the mirrors must be equal to half of the angle \( \alpha \), and this equality will not change after rotating the platform: the angle \( \alpha \) will remain the same, and hence the angle between the mirrors, equal to 15 degrees, must be maintained.

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