
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
Given an ellipse $\frac{x^2}{4} + y^2 = 1$ with its left and right foci labeled as $F_1$ and $F_2$ respectively, let $M$ be a point on the ellipse that is different from the endpoints of the major axis. Suppose $\angle F_1MF_2 = 2\theta$. If $I$ is the incenter of $\triangle MF_1F_2$, find the value of $|MI|\cos\theta$.

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